post-cracking characteristics of high performance fiber reinforced

POST-CRACKING CHARACTERISTICS OF HIGH PERFORMANCE FIBER REINFORCED CEMENTITIOUS

COMPOSITES

by

Supat W. Suwannakarn

A dissertation submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy (Civil Engineering)

in The University of Michigan 2009

Doctoral Committee:

Professor Sherif El-Tawil, Co-Chair Professor Emeritus Antoine E. Naaman, Co-Chair Professor Richard E. Robertson Associate Professor Gustavo J. Parra-Montesinos

© Supat W. Suwannakarn All Rights Reserved

2009

To My Family

`

ii

ACKNOWLEDGEMENTS

I wish to express my deepest gratitude and sincere appreciation to Professor

Antoine E. Naaman, and Professor Sherif El-tawil advisors and Co-Chair of my doctoral

committee, for their guidance, support and patience. I also would like to thank the

members of my doctoral committee, Associate Professor Gustavo Parra-Montesinos, and

Professor Richard E Robertson, for reviewing this thesis, and giving valuable comments

on my research work.

Special thanks to my colleagues, for their friendship and invaluable help. Thanks

are also due to laboratory technician Robert Fischer, Jan Pantolin, and Bob Spence, for

fabricating the test setup and specimen molds for my experiments.

Finally, I would like to dedicate this work to my family, who always give me their

love, care and encouragement for which I am extremely grateful.

iii

TABLE OF CONTENTS

DEDICATION …………………………...……………………………………… ii

ACKNOWLEDGEMENTS ………….……………………………….………… iii

LIST OF FIGURES …………………………………………………………...… x

LIST OF TABLES ………………………………………………………….…… xxiv

ABSTRACT ………………………………………………………..……………. xxviii

CHAPTER

1. INTRODUCTION ………………………………………………..…... 1

1.1 General ……………………………………...…………..…….. 1

1.2 Dissertation Objectives and Scope ……...………..……….…. 3

1.3 Research Significance ………………………..……..………... 4

1.4 Structure of the Dissertation ……...……………...……..…… 4

2. LITERATURE REVIEW MODELING OF HPFRCC IN

TENSION ………………………………………………...…………… 8

2.1 Introduction ………………………………………………...… 8

2.2 First Cracking Point (A), (σcc, εcc) …………………………… 9

2.3 Multiple Cracking Stage (A-B), (σcc, εcc to σpc, εpc) …...… 10

2.4 Maximum Tensile Strength Point (B), (σpc, εpc) ………...… 13

2.5 Localization Stage (B-C), (After σpc, εpc) …………..……… 16

2.6 Crack-Opening in HPFRCC ………………………………… 17

2.7 Crack Spacing in HPFRCC …...………………..…………… 21

3. EXPERIMENTAL PROGRAM FOR EVALUATING HPFRCC

TENSILE BEHAVIOR ……………………………………………… 25

3.1 Introduction ………………………………………………...… 25

3.2 Materials ……………………………………………………… 26

iv

3.2.1 Mortar ………………….…………………………… 26

3.2.2 Fibers ……………………………………………...… 27

3.2.2.1 Polyvinyl Alcohol (PVA) ………………… 29

3.2.2.2 Spectra ………………………………...…. 30

3.2.2.3 Hooked ……………………………….…… 31

3.2.2.4 Twisted Polygonal Steel Fibers (Torex) … 33

3.3 Specimen Preparation ……………………………………..… 34

3.4 Direct Tensile Test (Dogbone Test) …………………….…… 35

3.5 Crack Opening Displacement Test (COD, Notched

Prism Test) …………………………………………………… 38

3.6 Ring Tensile Test ……………………………………..……… 40

3.6.1 Result of Finite Element Analysis ………………… 42

3.7 Data Acquisition System. ……………………….…………… 45

3.8 Image Acquisition …………………..…………………...…… 47

3.9 Data Processing ……………………………………………… 48

3.10 Concluding Remarks ……………………………….……… 49

4. DIRECT TENSILE TESTS OF FIBER REINFORCED

CEMENT COMPOSITES………………………..……………...… 51

4.1 Introduction ……………………………………………......… 51

4.2 Direct Tensile Behavior of Mortar without Fibers ……...… 54

4.3 Direct Tensile Behavior of FRCC Reinforced PVA Fibers 60

4.3.1 Result of Test Series with PVA-H Fiber .………… 61

4.3.2 Result of Test Series with PVA-L Fiber .………… 66

4.3.3 Concluding Remarks ……………………………… 69

4.4 Direct Tensile Behavior of HPFRCC Reinforced Spectra

Fibers…………………………………………………………. 70


4.5 Direct Tensile Behavior of HPFRCC Reinforced with

Hooked Steel Fiber ………………………………………….. 79

4.5.1 Result of Test Series with High Strength Hooked

Steel Fiber .……………………………………….… 80

v

4.5.2 Result of Test Series with Regular Strength Hooked

Steel Fiber .……………………………………….… 88


4.6 Direct Tensile Behavior of HPFRCC Reinforced with Torex

Twisted Steel Fiber ………………………………………….. 91

4.6.1 Result of Test Series with High Torex Steel Fiber 92

4.6.2 Result of Test Series with Regular Strength Torex

Steel Fiber ……………………………...………...… 100


4.7 Comparison Between HPFRCC with Different Fibers ....… 103

4.8 Concluding Remarks on the Direct Tensile Tests of FRC

Composites with Different Fibers……………………..…… 104

5. STRESS VERSUS CRACK OPENING DISPLACEMENT TESTS 106

5.1 Introduction ………………………………………………..… 106

5.1.1 Typical Overall (σ-COD) Response ……………… 107

5.2 Typical Tensile Response (σ-COD) of Double-Notched

Specimens ……………………………………………………. 109

5.3 Notched Mortar Specimens without Fiber …………………. 114

5.4 Tensile Response (σ-COD) of Double-Notched Specimens

with PVA Fiber ……………………………………………… 116


with Spectra Fiber …………………………………………… 125


with High Strength Hooked Fiber …………..……………… 134


with Torex Fiber …………………………...………………… 142

5.8 Comparison of Test Series Reinforced with Difference

Fibers ………………………………………………………… 151

5.9 Concluding Remarks …………………..……………………. 152

vi

6. PROPOSED MODEL FOR POST-CRACKING BEHAVIOR OF

HPFRCC UNDER TENSION …...…………………………..……… 154

6.1 Introduction ………………………..………………………… 154

6.2 First Cracking Stress ……………………………………...… 156

6.2.1 FRC Reinforced with PVA Fiber ………………… 157

6.2.2 HPFRCC Reinforced with Spectra Fiber ……….. 158

6.2.3 HPFRCC Reinforced with Hooked Fiber ……….. 159

6.2.4 HPFRCC Reinforced with Torex Fiber …...…….. 160

6.3 Strain at First Cracking Stress …..………………………… 161

6.4 Maximum Post-Cracking Stress or Ultimate Stress ……… 162





6.4.5 Comparison of Coefficient λpc for Different Fibers 167

6.5 Multiple Cracking Behavior and Strain at Maximum

Stress …………………………………………………………. 168

6.5.1 Test Series with Hooked Fiber ……………………. 169

6.5.1.1 Calculation of Strain at Maximum Stress

for Test Series with Hooked Fiber ……… 172

6.5.2 Test Series with Torex Fiber ……...………………. 173


for Test Series with Torex Fiber ……...… 175

6.5.3 Test Series with Spectra Fiber ……………………. 176


for Test Series with Spectra Fiber ……… 178

6.6 Modeling the Softening Response after Localization ……… 179

6.7 Correlation between Direct Tensile Tests and Stress versus

Crack Opening Displacement Tests ……………………….. 186



vii



6.8 Outline of Stress-Strain Computations ……………………. 201

6.9 Verification of the Model ……………………………………. 203

6.9.1 Tensile Specimen with Torex Fiber ………………. 203

6.9.2 Other Model Predictions ………………………….. 205

6.10 Concluding Remarks ………………….…………………… 212

7. TENSILE VARIABILITY OF FIBER REINFORCED

CEMENTITIOUS MATERIALS …...……………………………… 214

7.1 Introduction ………………………………………..………… 214

7.2 Confidence Interval (CI) ……………………..……………… 218

7.3 Coefficient of Variation (COV) Analysis …...……………… 220

7.4 Goodness-of-Fit ………………………………………………. 223

7.5 Variability Graph ………….………………………………… 245

7.6 Concluding Remarks ……………………………………...… 248

8. RING-TENSILE TEST ……………………………………………… 250

8.1 Introduction …………………………………..……………… 250

8.2 Measurement Comparisons ………………………………… 251

8.3 Testing Results ………………………………………………. 258

8.4 Concluding Remarks ………………………………………… 262

9. CONCLUSION………………………………………………..……… 264

9.1 Summary …………………………………………………….. 264

9.2 Conclusions Drawn From Direct Tensile Test (Dogbone)

Test of HPFRCC ……………..........................……………... 265

9.2.1 PVA Fiber ………………………………………….. 265

9.2.2 Spectra Fiber ………………………………………. 266

9.2.3 Hooked Steel Fiber ………………………………… 266

9.2.4 Torex Steel Fiber …………………………………... 267

9.2.5 Direct Tensile Tests: General Conclusions ………. 267

9.3 Conclusions Drawn from the Stress-Crack Opening

Displacement (COD) Tests on Notched Tensile Prisms ....… 269

viii

9.3.1 PVA Fiber ………………………………...………... 269

9.3.2 Spectra Fiber ……………………………………….. 269

9.3.3 Hooked Steel Fiber …………………………………. 270

9.3.4 Torex Steel Fiber …………………………………… 270

9.3.5 Stress Crack Opening Displacement (COD)

Tests: General Conclusions ……………………….. 271

9.4 Conclusions from the Study on Modeling Tensile

Stress-Strain Response …………………………………….. 272

9.5 Conclusions from the Statistical Analysis and

Variability of Results ………………………………………… 273

9.6 Conclusions for the Ring Tensile Test Study ………………. 274

9.7 Main Conclusions ……………………………………….…… 274

9.8 Recommendation for Future Works ………………….……. 276

APPENDICES....……………………………………………………………….…… 277

APPENDIX A DIRECT TENSILE TESTS USING SIFCON……….….278

APPENDIX B DEFINITION OF FIRST CRACKING, MAXIMUM

STRESS POINT, AND LOCALIZATION STARTING

POINT……………………………………………….…….282

APPENDIX C VARIABILITY OBSERVED IN SAME TEST SERIES

USING 12 SPECIMENS PREPARED ON THE SAME

DAY AND TESTED ON THE SAME DAY………..…...284

APPENDIX D COMPARISON OF α1α2, α1α2τ, λpc AND λpcτ……..287

BIBLIOGRAPHY………………………………………………………………...… 289

ix

LIST OF FIGURES

FIGURE

1.1 Stress-strain relation of typical tensile behavior of HPFRCC………........ 6

1.2 Typical schematic tensile behavior of HPFRCC, i.e., strain hardening

behavior………........................................................................……………… 6

1.3 Research plan and objectives………………………………………...…….. 7

2.1 HPFRCC stress-strain relation………….……………….………..……….. 8

2.2 Crack model for HPFRCCs showing (a) fracture zone and (b) possible

Stress distribution…………………….….………….……………....……... 24

3.1 Flow chart of experimental program……….…………………………..… 26

3.2 Fibers used in present research, (a) PVA, (b) Spectra, (c) Hooked,

and (d) Torex…………………………………………………..…..………. 28

3.3 Fibers used in present research: length comparison ....……….………… 28

3.4 PVA fiber……………………………………………………….….……..… 30

3.5 PVA fiber in composites………………………………………….…......… 30

3.6 Spectra fiber……….………………………………………………..……… 31

3.7 Spectra fiber in composite………………………………………….……… 31

3.8 Hooked steel fiber (before mixing)………...………………………………. 32

3.9 Hooked steel fiber in composites………………………………….………. 32

3.10 Pull-out mechanism in Hooked steel fiber………………………………… 32

3.11 Formation of two plastic hinges for maximum resistance……….…….… 32

3.12 Torex fiber geometry………………….…………………………………… 33

3.13 Torex steel fiber……………………………………………………………. 34

3.14 Torex steel fiber in composites………………………………………….… 34

x

3.15 Specimen dimension of tensile dogbone specimens (mm) ……………..… 36

3.16 Specimen dimension of tensile dogbone specimens (inch) …………….… 36

3.17 Tensile test setup, (a) specimen ready for testing and

(b) instrumentation………………..……………………………………….. 37

3.18 Dogbone specimen……………………………………………………......… 37

3.19 Mold for dogbone specimen and added end reinforcement…………..…. 37

3.20 Dimensions of notched tensile prim for crack-opening displacement test

(mm) ……………………………………………………….……………….. 38

3.21 Dimensions of notched tensile prism for crack-opening displacement

test (inch) ………………………………………………………………..…. 39

3.22 Crack opening displacement test set-up………………………………...… 40

3.23 Test setup configuration………………… ………………………………… 40

3.24 Ring tensile test set-up…….….…………………………………..………… 41

3.25 HPFRCC ring specimen………………………...…………………………. 41

3.26 Ring HPFRCC specimen……..…………………….….…………………… 42

3.27 Ring HPFRCC setup………….………………..…………………………… 42

3.28 Interaction between steel plate and cone wedge setup………..…….……. 42

3.29 Stress-contour of finite element analysis for two, four, and eight

piece steel-plate configurations……………..……………………………… 42

3.30 Specimen setup………………………………………………………….….. 43

3.31 Optotrak sensors…………………………………………………………… 43

3.32 Testing machine and tensile ring setup…….……………………………… 44

3.33 Testing machine and tensile setup……………………………………….… 44

3.34 LVDT setup at specimen’s surface……………...……………………….…44

3.35 LVDT setup at cone wedge…………………..…………..…………….……44

3.36 Data collecting system for the direct tension tests....……………………...45

3.37 Data acquisition system……………………………….………………….…46

3.38 View of real-time records of stress-elongation response curves…..…….. 46

3.39 Diagram for image acquisition system……………….…………………… 47

3.40 Image acquisition processing hardware (NI-CVS1453)…………………. 48

3.41 Image acquisition system…………………………………………………... 48

xi

3.42 Typical data and average curve…………...……………………..…….…. 48

4.1 Typical tensile behavior of (a) strain hardening composite and

(b) strain softening composite………..……………………...…………….. 52

4.2 Typical behavior of HPFRCC……………………………………………... 53

4.3 Strain at maximum stress and strain at the end of multiple cracking

stages…..……………………………………………………………………. 54

4.4 Testing results of specimen without fiber (control specimen)…………… 58

4.5 Testing of typical specimen without fiber (D-N-O-7), (a) initial status,

(b) maximum status, and (c) failure ………………….………..…………. 59

4.6 Dogbone specimen without fibers (specimen 1, 2, 3)….…….…………… 59

4.7 Dogbone specimen without fibers (specimen 4, 5, 6)……………………... 60

4.8 Stress-strain curves of specimens reinforced with PVA-H (D-P-H),

(a) Vf = 0.75%, (b) Vf = 1.0%, (c) Vf = 1.5%, and (d) Vf = 2.0% ……….. 62

4.9 Crack in direct tensile specimen reinforced with PVA fiber…………….. 63

4.10 Comparison of average curves for specimens reinforced with

PVA-H fiber with different volume fractions…………………………… ...63

4.11 Comparison of different properties of test series with oiled PVA

fibers at different fiber volume fractions, (a) first cracking stress,

(b) energy at first cracking stress, (c) maximum stress, and (d) energy

at post cracking stress ………………………………………………..……. 64

4.12 Specimens reinforced with PVA-H after testing (D-P-H) ……………….. 66

4.13 Comparison of tensile response of specimens with PVA-L fiber at

different fiber volume fraction……………………..……………………… 66

4.14 Typical failure sections of specimens reinforced with (a) non-oiled

PVA-L fibers and (b) oiled PVA-H fibers………………………………… 67

4.15 Stress-strain curves of specimens reinforced with Spectra (D-S) fiber

at volume fractions of: (a) 0.75%, (b) 1%, (c) 1.5%, and (d) 2% …...… ...71

4.16 Comparison of average curves for specimen reinforced with Spectra

fiber at different fiber volume fractions………..……….………………… 72

4.17 First cracking stress versus volume fraction………..…………………….. 73

4.18 Strain at first cracking stress versus volume fraction…………..………... 73

xii

4.19 Energy at first cracking stress versus volume fraction……….……...…... 73

4.20 Specimen at maximum stress……………………………………………… 74

4.21 Strain at maximum stress……………………………………………….… 74

4.22 Specimens’ energy at maximum stress………………………….………… 75

4.23 Variation of strain at the end of multiple cracking…………….…………. 75

4.24 Variation of energy at the end of multiple cracking………………...…….. 75

4.25 Stress strain curves of HPFRCC reinforced with high strength

Hooked steel fiber at different volume fractions of fiber and average

curves, (a) Vf = 0.75%, (b) Vf =1.0%, (c) Vf =1.5%, and (d) Vf =2.0%..... 81

4.26 Average stress strain curves of HPFRCC reinforced with high strength

Hooked steel fiber………………………………………………………….. 82

4.27 Stress at first cracking versus fiber volume fraction ...……………...….. 83

4.28 Strain at first cracking versus fiber volume fraction….………………… 83

4.29 Energy at first cracking versus fiber volume fraction……...………..….. 83

4.30 Maximum stress versus fiber volume fraction……………..…………….. 84

4.31 Strain at maximum stress versus fiber volume fraction………………… 84

4.32 Energy at maximum stress point versus fiber volume fraction……...….. 84

4.33 Strain at the end of multiple cracking versus fiber volume fraction……. 85

4.34 Energy at the end of multiple cracking versus fiber volume fraction….. 85

4.35 Average stress strain curves of HPFRCC reinforced with regular

strength Hooked steel fiber at different volume fractions……………..... 88

4.36 Stress strain curves of HPFRCC reinforced with high strength

Torex steel fiber at different volume fractions of fiber and average

curves, (a) Vf = 0.75%, (b) Vf =1.0%, (c) Vf =1.5%, and (d) Vf =2.0%..... 93

4.37 Average stress strain curves of HPFRCC reinforced with high strength

Torex steel fiber at different fiber volume fractions…………………….. 94

4.38 First cracking stress versus fiber volume fraction……………………….. 95

4.39 Strain at first cracking stress versus fiber volume fraction……...…..….. 95

4.40 Energy at first cracking stress versus fiber volume fraction…….…..….. 95

4.41 Maximum stress versus fiber volume fraction ……………….………….. 96

4.42 Strain at maximum stress versus fiber volume fraction……………..….. 96

xiii

4.43 Energy at maximum stress versus fiber volume fraction…………….….. 96

4.44 Strain at the end of multiple cracking stage versus fiber volume

fraction…....………………..……………………………………………….. 97

4.45 Energy at the end of multiple cracking stage versus fiber volume

fraction…………………………………………………………………..….. 97

4.46 Average stress strain curves of HPFRCC reinforced with regular

strength Torex steel fiber at different fiber volume fractions……….… 100

4.47 Comparison of average stress-strain response of FRCC test series with

different fibers at fiber volume fractions of: a) Vf = 0.75%, (b) Vf =1.0%,

(c) Vf =1.5%, and (d) Vf =2.0%………….…………………………….… 103

5.1 Typical behavior of notch HPFRCCs test…………...…………………. 107

5.2 Typical behavior of notched specimen, (a) with one crack and (b) with

several cracks…………………………………………………………...… 108

5.3 Zone of cracking influence in notch specimen……………………….… 108

5.4 Typical crack propagation and localization in HPFRCC…………....… 109

5.5 (a) Strain hardening material with multiple cracks and (b)

corresponding specimen……………………………………………..…... 110

5.6 (a) Strain hardening material with single major crack and (b)

corresponding specimen…………………………………………………. 111

5.7 (a) Strain softening material with a post cracking stress that picks up

after a first dip and (b) corresponding specimen……………………..… 112

5.8 (a) Strain softening material and (b) corresponding specimen…………113

5.9 Typical stress-crack opening displacement curves…………………….. 113

5.10 Stress-displacement curves of notched specimens with 2% Spectra

fiber and average curve………………………………………………...… 114

5.11 Stress-displacement curve of plain mortar matrix without fiber……… 115

5.12 (a) (σ-COD) curves of specimens reinforced with 0.75% PVA fiber

and (b) photo of tested specimens reinforced with 0.75% PVA fiber…. 117

5.13 (a) (σ-COD) curves of specimens reinforced with 1% PVA fiber and

(b) photo of tested specimens reinforced with 1% PVA fiber…….…… 117

xiv

5.14 (a) (σ-COD) curves of specimens reinforced with 1.5% PVA fiber and

(b) photo of tested specimens reinforced with 1.5% PVA fiber ….…… 118

5.15 (a) (σ-COD) curves of specimens reinforced with 2% PVA fiber and

(b) photo of tested specimens reinforced with 2% PVA fiber ………… 118

5.16 Comparison of average (σ-COD) curves with PVA fiber……………… 119

5.17 Definition of localization start point, (a) for specimen with only one

global maximum stress and (b) for specimen with a second local

maximum ………………………………………………………………… 120

5.18 First cracking stress……………………………………………………… 122

5.19 Displacement at first cracking stress…………………………………… 122

5.20 Energy at first cracking stress…………………………………………… 122

5.21 Maximum stress ………………………………………………………….. 123

5.22 Displacement at maximum stress………………………………………... 123

5.23 Energy at maximum stress…………………………………………….… 123

5.24 Displacement at localization start……………………………………...…124

5.25 Energy at localization start…………………………………………….… 124

5.26 (a) (σ-COD) curves of specimens reinforced with 0.75% Spectra and

(b) photo of tested specimens reinforced with 0.75% Spectra ………… 126

5.27 (a) (σ-COD) curves of specimens reinforced with 1% Spectra and

(b) photo of tested specimens reinforced with 1% Spectra……….…… 126

5.28 (a) (σ-COD) curves of specimens reinforced with 1.5% Spectra and

(b) photo of tested specimens reinforced with 1.5% Spectra…………... 127

5.29 (a) (σ-COD) curves of specimens reinforced with 2% Spectra and

(b) photo of tested specimens reinforced with 2% Spectra ………….… 127

5.30 Comparison of average (σ-COD) curves with Spectra fiber………...… 128

5.31 First cracking stress…………………………………..………………..… 129

5.32 Displacement at first cracking stress……………………………………. 129

5.33 Energy at first cracking stress…………………………………………... 129

5.34 Maximum stress…………………………………………………………... 130

5.35 Displacement at maximum stress……………………………………….. 130

5.36 Energy at maximum stress………………………………………………. 130

xv

5.37 Typical multiple cracking in notched specimens with Spectra fiber..... 131

5.38 Displacement at the end of multiple cracking stage…………………… 131

5.39 Energy at the end of multiple cracking stage…………………………... 131

5.40 (a) (σ-COD) curves of specimens reinforced with 0.75% Hooked fiber

and (b) photo of tested specimens reinforced with 0.75% Hooked fiber 135

5.41 (a) (σ-COD) curves of specimens reinforced with 1% Hooked fiber

and (b) photo of tested specimens reinforced with 1% Hooked fiber … 135

5.42 (σ-COD) curves of specimens reinforced with 1.5% Hooked fiber …… 136

5.43 (σ-COD) curves of specimens reinforced with 2% Hooked fiber ……... 136

5.44 Comparison of average (σ-COD) curves with Hooked fiber ………...…136

5.45 First cracking stress……………………………..……………………….. 137

5.46 Displacement at first cracking stress………………………………….… 137


5.48 Maximum stress…………………………………………………………... 138

5.49 Displacement at maximum stress………………………………………... 138


5.51 Displacement at the end of multiple cracking stage……………………. 139

5.52 Energy at the end of multiple cracking stage…………………………… 139

5.53 (a) (σ-COD) curves of specimens reinforced with 0.75% Torex fiber

and (b) photo of tested specimens reinforced with 0.75% Torex fiber…143

5.54 (a) (σ-COD) curves of specimens reinforced with 1% Torex fiber

and (b) photo of tested specimens reinforced with 1% Torex fiber…… 143

5.55 (a) (σ-COD) curves of specimens reinforced with 1.5% Torex fiber

and (b) photo of tested specimens reinforced with 1.5% Torex fiber .…144

5.56 (a) (σ-COD) curves of specimens reinforced with 2% Torex fiber

and (b) photo of tested specimens reinforced with 2% Torex fiber ...… 144

5.57 Comparison of average (σ-COD) curves with Torex fiber…………...…145

5.58 First cracking stress……………………………………..……………….. 146

5.59 Displacement at first cracking stress………………….………………… 146


xvi

5.61 Maximum stress………………………………..……………………..…... 147

5.62 Displacement at maximum stress……………………………..………… 147


5.64 Displacement at the end of multiple cracking stage ……..……………. 148

5.65 Energy at the end of multiple cracking stage……………………………148

5.66 Comparison of stress-crack opening displacement for the four fibers

used at (a) Vf = 0.75%, (b) Vf =1.0%, (c) Vf =1.5%, and (d) Vf =2.0% .. 151

6.1 Proposed diagram of HPFRCC model…………………….………….… 154

6.2 Typical tensile response of HPFRCC and chain material……………… 155

6.3 Variation of first cracking stress for test series with PVA fiber………. 158

6.4 Product α1α2 versus Vf…………………………………………………… 158

6.5 Variation of first cracking stress for test series with Spectra fiber….… 159

6.6 Product α1α2 versus Vf……………………………………………………. 159

6.7 Variation of first cracking stress for test series with Hooked fiber…... 160

6.8 Product α1α2 versus Vf …………………………………………………… 160

6.9 Variation of first cracking stress for test series with Torex fiber …...… 161

6.10 Product α1α2 versus Vf …………………………………………………… 161

6.11 Variation of maximum stress for test series with PVA fiber…………… 163

6.12 Coefficient λpc versus Vf………………………………………………….. 163

6.13 Variation of maximum stress for test series with Spectra fiber ………. 164

6.14 Coefficient λpc versus Vf ………………………………………………… 164

6.15 Variation of maximum stress for test series with Hooked fiber …….… 166

6.16 Coefficient λpc versus Vf ……………………………………………….… 166

6.17 Variation of maximum stress for test series with Torex fiber ………… 167

6.18 Coefficient λpc versus Vf ………………………………………………… 167

6.19 λpc of HPFRCC and Vf…………………………………………………… 168

6.20 Typical increase in number of cracks with fiber volume fraction…..… 169

6.21 Crack formation under increasing load in test series reinforced with

1.5% Hooked steel ……………………………..……………………….…169

6.22 Average crack spacing at saturation for test series with Hooked fiber.. 170

6.23 Average crack width at saturation for test series with Hooked fiber … 170

xvii

6.24 Average crack width versus average crack spacing for test series

with Hooked fiber ……………………………………………………...… 171

6.25 Crack formation under increasing load in test series reinforced

with 1.5% Torex steel fiber ……………………………………..……….. 173

6.26 Average crack spacing at saturation for test series with Torex fiber … 174

6.27 Average crack width at saturation for test series with Torex fiber ……174

6.28 Average crack width versus average crack spacing at saturation in tensile

test series with Torex fiber ……………………………….……………… 175

6.29 Average crack spacing at saturation for test series with Spectra fiber ..176

6.30 Average crack width at saturation for test series with Spectra fiber ..... 176

6.31 Average crack width versus average crack spacing at saturation in

tensile test series with Spectra fiber……………………………………... 177

6.32 Typical tensile specimens reinforced with Spectra fiber at the end of

test …………………………………………………………………..…….. 178

6.33 Stress-displacement relation after localization ……………………….... 179

6.34 Localization-failure in test series with PVA fiber …..…………..……… 180

6.35 Fibers assumed to pull-out for specimens with Hooked, Torex,

and Spectra fibers………………………………………………………… 180

6.36 Embedded length of aligned fibers bridging a crack………………...… 182

6.37 Example of normalized stress-displacement response of test series

reinforced with 2% Torex fiber…………………………………………. 184

6.38 Normalized curves after localization and their average for the test

series with: (a) PVA fiber, (b) Spectra fiber, (c) Hooked fiber, and

(d) Torex fiber ……………………………………………………….….... 185

6.39 Predicted average stress-displacement curves after localization for

all test series with notched specimens…………………………………... 186

6.40 Correlation between strain and maximum stress from a direct tensile

test and displacement at maximum stress from a notched prism test… 186

6.41 Zone of cracking influence around main crack………………………… 187

6.42 Correlation between “zone of cracking” from a stress-crack

opening displacement test and a direct tensile test (Dogbone test)……. 187

xviii

6.43 Schematic stress-strain curve and crack clusters………………………. 188

6.44 Definition of Ede ,Ene, Dde, and Dne………………………………………. 189

6.45 Typical failure crack in a tensile specimen reinforced with PVA fiber...190

6.46 Cracking in a notched prism with PVA fiber…………………….…….. 190

6.47 Ede/Ene for test series with PVA fiber…………………………………..…191

6.48 Typical failure crack in a tensile specimen reinforced with Spectra

fiber test ……............................................................................................... 192

6.49 Cracking in a notched prism with Spectra fiber ………...……………... 192

6.50 Number of equivalent cracks in a cluster in notched test series with

Spectra fiber……………………………..……………………………….. 193

6.51 Ede/Ene for test series with Spectra fiber………………………………… 193

6.52 Comparison of the number of cracks in direct tensile test series with

Spectra fiber (Dogbone test) obtained experimentally and estimated

from the energy method………………………………………………….. 194

6.53 Typical failure crack in a tensile specimen reinforced with Hooked

fiber……………………………………………………………………….. 195

6.54 Cracking in a notched prism with Hooked fiber ………………………. 195


Hooked fiber …………………………………..…………………………. 196

6.56 Ede/Ene for test series with Hooked fiber …………………………………197


Hooked fiber (Dogbone test) obtained experimentally and estimated

from the energy method………………………………………………..… 197

6.58 Typical failure crack in a tensile specimen reinforced with Torex

fiber………………………………………………………………………... 198

6.59 Cracking in a notched prism with Torex fiber ………………………….198


Torex fiber …………………....................................................................... 199

6.61 Ede/Ene for test series with Torex fiber ………………………………..… 199

xix


Torex fiber (Dogbone test) obtained experimentally and estimated

from the energy method………………………………………….…….… 200

6.63 Flowchart of proposed post-cracking model…………………………… 202

6.64 Comparison of predicted versus experimental stress-strain curve……. 205

6.65 Schematic stress-strain response of FRC reinforced with PVA fiber

(Direct tensile test) ……………………………………………..……….... 206

6.66 Schematic stress-displacement response of FRC reinforced with PVA

fiber (Notch tensile test)…………………………………………………... 206

6.67 Schematic stress-strain response of FRC reinforced with Spectra fiber

(Direct tensile test)………………………………………………………... 207

6.68 Schematic stress-strain response of FRC reinforced with Hooked fiber

(Direct tensile test)………………………………………………………... 207

6.69 Schematic stress-displacement response of FRC reinforced with Hooked

fiber (Notch tensile test)….……………………………………………….. 208

6.70 Schematic stress-strain response of FRC reinforced with Torex fiber

(Direct tensile test)…..…………………………… ……………………… 208

6.71 Schematic stress-displacement response of FRC reinforced with

Torex fiber (Notch tensile test)……………………………………….….. 209

6.72 Crack formation under increasing load in test series reinforced

with 0.75% Torex steel fiber (Notch test) ……………………………… 213

7.1 Probability density function with different μ and σ …………………... 216

7.2 Cumulative distribution function………………………………………. 216

7.3 95% confidence interval in normal distribution ……………………..… 216

7.4 Energy at localization start: area under the curve, toughness …………217

7.5 Multiple cracking stage in notch HPFRCC reinforced Spectra fiber … 217

7.6 (a) Direct tensile curve of HPFRCC reinforced Torex 1.5% and

(b) its corresponding confidence interval ………………………………. 218

7.7 (a) Direct tensile curve of HPFRCC reinforced Torex 2.0% and

(b) its corresponding confidence interval ………………………..………219

xx

7.8 (a) Direct tensile curve of HPFRCC reinforced Hooked 1.5% and


7.9 (a) Direct tensile curve of HPFRCC reinforced Hooked 2.0% and


7.10 Coefficient of variation, maximum stress ………………………………. 221

7.11 Coefficient of variation, energy at maximum stress...………………….. 221

7.12 Coefficient of variation, strain at maximum stress ……………………. 222

7.13 Comparison of coefficient of variation with different fiber types…....... 223

7.14 (a) Distribution of maximum stress, (b) cumulative histogram, and

(c) normal probability plot for HPFRCC reinforced Spectra 1.5%....... 227

7.15 (a) Distribution of energy at the end of multiple cracking stage, (b)

cumulative histogram, and (c) normal probability plot for HPFRCC

reinforced Spectra 1.5%…………………………………………………. 228

7.16 (a) Distribution of strain at the end of multiple cracking stage,

(b) cumulative histogram, and (c) normal probability plot for

HPFRCC reinforced Spectra 1.5%…………………………………...…. 229

7.17 (a) Distribution of maximum stress, (b) cumulative histogram,

and (c) normal probability plot for HPFRCC reinforced Spectra

2.0%.............................................................................................................. 230

7.18 (a) Distribution of energy at the end of multiple cracking stage, (b)

cumulative histogram, and (c) normal probability plot for HPFRCC

reinforced Spectra 2.0%…………………………………………………. 231

7.19 (a) Distribution of strain at the end of multiple cracking stage,

(b) cumulative histogram, and (c) normal probability plot for

HPFRCC reinforced Spectra 2.0%……………………….……………. 232

7.20 (a) Distribution of maximum stress, (b) cumulative histogram, and (c)

normal probability plot for HPFRCC reinforced Hooked 1.5%……… 233

7.21 (a) Distribution of energy at maximum stress, (b) cumulative

histogram, and (c) normal probability plot for HPFRCC reinforced

Hooked 1.5%……………………………………………………………... 234

xxi

7.22 (a) Distribution of strain at maximum stress, (b) cumulative


Hooked 1.5%…………………………………………………………….. 235

7.23 (a) Distribution of maximum stress, (b) cumulative histogram,

and (c) normal probability plot for HPFRCC reinforced Hooked

2.0%.............................................................................................................. 236



Hooked 2.0%…………………………………………………………….. 237



Hooked 2.0%………………………………………………….………….. 238


(c) normal probability plot for HPFRCC reinforced Torex 1.5%……. 239



Torex 1.5%................................................................................................... 240



Torex 1.5%……………………………………………………………..… 241


(c) normal probability plot for HPFRCC reinforced Torex 2.0%…..… 242



Torex 2.0%………………………………………………………….….… 243



Torex 2.0%…………………………………………………..……………. 244

7.32 First cracking stress range and average ……………………………….. 245

7.33 Maximum stress range and average ……………………………………. 246

7.34 Strain at maximum stress range and average ………………………..… 247

xxii

7.35 Energy at maximum stress range and average ………………………… 248

8.1 Comparison of specimens between ring-tensile and Dogbone shaped… 251

8.2 Illustrates the displacement obtained (a) from the Optotrak, (b) the

LVDT is on the perimeter of the specimen’s surface, and (c) at the

tip of the cone wedge………………………………………………..…... 251

8.3 An Optotrak displacement sensor location…………………………….. 252

8.4 Ring-tensile test results of Hooked 2.0% specimen 2 composites with

varying measurements………................................................................... 253

8.5 Ring-tensile test results of Hooked 1.0% composites with various types

of measurements………………………………………………………….. 254

8.6 Localization crack of PVA specimen……………………………………. 255

8.7 Localization crack of Hooked specimen………………………………… 256

8.8 Multiple cracking from inside surface of ring specimen…..................... 256

8.9 One of the four minor localization cracks………………………………. 257

8.10 Multiple cracking for both inside and outside surface of specimens….. 257

8.11 Testing result of all ring specimens tested…………………………….… 258

8.12 Comparison of maximum stress of all ring-tensile specimens tested….. 259

8.13 Cracking steps in ring-tensile specimen (a) elastic stage, (b) multiple

cracking stage, and (c) localization……………………………………… 260

8.14 Stress-strain result of PVA 2% specimen with error in strain hardening and multiple cracking behavior…………………..…………. 262

xxiii

LIST OF TABLES

TABLE

3.1 Cement compositions and properties of Portland cement type III.....… 27

3.2 Mix proportion of matrix by weight…………………………………...…27

3.3 Fiber properties…………………………………………………………… 29

3.4 Summary of testing program…………………………………………… 50

4.1 Common mechanical properties of mortar…………………………….. 55

4.2 Identification of mortar specimens and mortar composition…………. 55

4.3 Direct tensile test of mortar (US-units)…………………………………. 57

4.4 Direct tensile test of mortar (SI-units)………………………………...… 57

4.5 Summary of direct tensile test series with PVA fiber……………………61

4.6 Test results of direct tensile specimens reinforced with PVA-H fiber

(US-units)…………………. ………………………………………………65

4.7 Test results of direct tensile specimens reinforced with PVA-H fiber

(SI-units)……………………………………… …………………………. 65

4.8 Summary of test results of specimens reinforced PVA-L (US-units) … 68

4.9 Summary of test results of specimens reinforced PVA-L (SI-units) ..… 69

4.10 Summary of HPFRCC reinforced Spectra test………………………….71

4.11 Volume fraction of fiber and the stress ratio…………………………….74

4.12 Summary of test results of specimens reinforced with Spectra fiber

(US-units)……………………………………………….. ……………….. 77

4.13 Summary of test results of specimens reinforced with Spectra fiber

(SI-units) ………………………………………………….………………. 78

xxiv

4.14 Identification of specimen reinforced Hooked fiber………………….... 80

4.15 Summary of test results of specimens reinforced with high strength

Hooked fiber (US-units) …………………………………………………. 86

4.16 Summary of test results of specimens reinforced with high strength

Hooked fiber (SI-units) …………………………………..……………… 87

4.17 Summary of test results of specimens reinforced with regular

strength Hooked steel fiber (US-units)…………………………………... 89

4.18 Summary of test results of specimens reinforced with regular strength

Hooked steel fiber (SI-units)………………………………………………. 90

4.19 Identification of specimen reinforced Torex fiber…………………….... 92

4.20 Summary of test results of specimen reinforced high strength

Torex fiber (US-units) ……………………………………………………. 98

4.21 Summary of test results of specimen reinforced high strength

Torex fiber (SI-units) …………………………………………………….. 99

4.22 Summary of test results of specimens reinforced regular strength

Torex fiber (US-units)………………………………………………….… 101

4.23 Summary of test results of specimens reinforced regular strength

Torex fiber (SI-units)……………………………………………..…..….. 102

5.1 Key results for notched specimens without fiber………………………. 116

5.2 Test series of notched specimens reinforced with PVA fiber………….. 116

5.3 Summary of test results for notched specimens reinforced with PVA

fiber (US-units)……………………………………………………………. 121

5.4 Summary of test results for notched specimens reinforced with PVA

fiber (SI-units)……..……………………………………………………... 121

5.5 Test series of notched specimens reinforced with Spectra fiber………. 125

5.6 Summary of test results for notched specimens reinforced with

Spectra fiber (US-units)…………………………………………………... 132


Spectra fiber (SI-units)……...……………………………………..……... 133

5.8 Test series of notched specimens reinforced with Hooked fiber……….. 134

xxv


Hooked fiber (US-units)…...……………………………………………... 140


Hooked fiber (SI-units)………………………………………………….... 141

5.11 Test series of notched specimen reinforced reinforced with Torex

fiber…………………………………………………………………..…… 142


Torex fiber (US-units) …………………………………………………… 149


Torex fiber (SI-units)…………………….……………………………….. 150

6.1 Average first cracking stress of tensile test series with PVA fiber ……. 158

6.2 Average first cracking stress of tensile test series with Spectra fiber ... 159

6.3 Average first cracking stress of tensile test series with Hooked fiber … 160

6.4 Average first cracking stress of tensile test series with Torex fiber ……161

6.5 Average post-cracking strength of tensile test series with PVA fiber…. 163

6.6 Average post-cracking strength of tensile test series with Spectra fiber 164

6.7 Average post-cracking strength of tensile test series with Hooked fiber 165

6.8 Average post-cracking strength of tensile test series with Torex fiber... 167

6.9 Series with Hooked, Average crack spacing, Average crack width and

predicted strain…………………………….………………………………172

6.10 Series with Torex, Average crack spacing, Average crack width and

predicted strain ……………………...………………………………...…. 176

6.11 Series with Spectra, Average crack spacing, Average crack width and

predicted strain ……………………………………………………….….. 179

6.12 Value of parameters K1 and K2 for (Eq. 6.27)…...……………………… 183

6.13 Surface energy and area under the curve obtained from tests

(US-units)………………………………………………………………..... 200

6.14 Surface energy and area under the curve obtained from tests

(SI-units) …………………………………………………………………. 201

6.15 Energy ratio and influence unit/length of crack-opening displacement

test…………………………………………………………………………. 201

xxvi

6.16 Computations of stresses and displacements for the example

specimen…………………………………………………………………… 204

6.17 Summary of computations for the stress-strain curves of HPFRCC

tensile specimens (a, b, c, d, e, f) ………………………………………… 209

7.1 Diagram for analysis of tensile histogram and normal probability

curve…………………………………………………………..…………... 218

7.2 Normality test results at 95% confidence level………………………… 224

7.3 Statistical parameters of HPFRCC specimens (US-units) ………….… 225

7.4 Statistical parameters of HPFRCC specimens (SI-units) …………….. 226

xxvii

ABSTRACT

Post-Cracking Characteristics of High Performance Fiber Reinforced Cementitious Composites

by

Supat W. Suwannakarn

Co-Chairs: Sherif El-Tawil and Antoine E. Naaman

The application of high performance fiber reinforced cement composites

(HPFRCC) in structural systems depends primarily on the material’s tensile response,

which is a direct function of fiber and matrix characteristics, the bond between them, and

the fiber content or volume fraction. The objective of this dissertation is to evaluate and

model the post-cracking behavior of HPFRCC. In particular, it focused on the influential

parameters controlling tensile behavior and the variability associated with them. The key

parameters considered include: the stress and strain at first cracking, the stress and strain

at maximum post-cracking, the shape of the stress-strain or stress-elongation response,

the multiple cracking process, the shape of the resistance curve after crack localization,

the energy associated with the multiple cracking process, and the stress versus crack

xxviii

opening response of a single crack. Both steel fibers and polymeric fibers, perceived to

have the greatest potential for current commercial applications, are considered. The main

variables covered include fiber type (Torex, Hooked, PVA, and Spectra) and fiber

volume fraction (ranging from 0.75% to 2.0%). An extensive experimental program is

carried out using direct tensile tests and stress-versus crack opening displacement tests on

notched tensile prisms. The key experimental results were analysed and modeled using

simple prediction equations which, combined with a composite mechanics approach,

allowed for predicting schematic simplified stress-strain and stress-displacement response

curves for use in structural modeling. The experimental data show that specimens

reinforced with Torex fibers performs best, follows by Hooked and Spectra fibers, then

PVA fibers. Significant variability in key parameters was observed througout suggesting

that variability must be studied further.. The new information obtained can be used as

input for material models for finite element analysis and can provide greater confidence

in using the HPFRC composites in structural applications. It also provides a good

foundation to integrate these composites in conventional structural analysis and design.

xxix

1

CHAPTER 1

INTRODUCTION

1.1 General

Cementitious materials, such as concrete and mortar, exhibit brittle tensile

behavior. However, their behavior can be significantly improved by adding discontinuous

fibers. Historically, Joseph Lambot’s (1849) fiber application reveals the idea of using

continuous fibers in mesh form to create new building materials, which led to the

development of ferrocement and reinforced concrete. Romualdi et al (1963) had proposed

using short randomly oriented fibers in order to improve the matrix’s brittle nature under

tensile loading. Today, several types of reinforcing fibers, in various shapes and sizes,

such as steel, polymer, glass, carbon, or natural fiber, are produced and used widely.

The main advantage of using discontinuous fibers in brittle matrices, such as a

cementitious matrix, is usually realized only after the matrix cracks. The fibers can

prevent a sudden loss in load-carrying capacity of the cracked composite by providing a

load transfer mechanism across the crack, resulting in a pseudo-ductile response.

When a load is applied to a fiber reinforced composite element, it is distributed to

both the matrix and the fibers. The transmission of forces between the fibers and the

matrix occurs through interfacial bond, defined as the bond shear stress at the interface

between the fiber and the surrounding matrix. The fibers contribute primarily to the post

cracking response of the composite by bridging the cracks and providing resistance to

2

crack opening.

According to Naaman and Reinhardt (1987, 1992, 1996), fiber reinforced cement

composites can be classified into the following two categories: conventional fiber

reinforced cementitious composite (FRCC) and high performance fiber reinforced

cementitious composite (HPFRCC), Fig. 1.1. More recently, however, and in order to

minimize confusion, Naaman and Reinhardt (2003) suggested the use of a broader

classification which apply to all fiber reinforced cement composites, namely: either

strain-softening or strain-hardening in tension after first cracking.

When the first crack occurs, FRCC response in tension softens, while in contrast,

HPFRCC response hardens. In other words, if the post cracking strength (σpc) is higher

than that at first cracking (σcc), then the composite is considered to be a high performance

material (or equivalently a strain-hardening material). An alternative classification based

on energy is possible when the energy needed to create a new crack is less than the

energy required to extend a former crack, the multiple cracking resulting from this

condition is considered characteristic of HPFRCC behavior.

Generally, the tensile stress-elongation response of HPFRCC can be classified

into three parts; the elastic stage, wherein the matrix is not cracked up to σcc, the multiple

cracking stage (σpc), and the damage localization stage, or crack opening stage (Fig. 1.2).

In the elastic stage, the composite exhibits linear behavior up to first cracking (σcc). After

first cracking, the multiple cracking stages lead to a strain hardening effect during with

the load still increases up to the ultimate strength (σpc). After the peak load, damage

localized failure occurs via a single critical crack opening. Thereafter, resistance

decreases with the opening of the critical crack; i.e. softening response take place

Fibers bridging a crack can absorb more or less energy depending on their bond

characteristics. The pull-out process involves first, a debonding action which provides an

alternative path for the crack to follow, and is preceded by the formation of a new surface

at the fiber matrix interface. Moreover, the fiber deformation and compliance during pull-

out contributes directly to the total deformation of the composite.

The behavior of FRC can be classified into three groups according to application,

fiber volume fraction and fiber effectiveness. Such classification leads to : 1) very low

volume fraction of fiber (<1%), which has been used for many years now such as for

3

early age plastic shrinkage control or pavement reinforcement, 2) moderate volume

fraction of fiber (1%-2%), such as used in both cast-in-place and structural members for

their improved modulus of rupture (MOR), fracture toughness, impact resistance and

other desirable mechanical properties, and 3) high volume fractions of fibers (more than

2%) for special applications such as impact and blast resistance structures; these include

SIFCON (Slurry Infiltrated Fiber Concrete), SIMCON (Slurry Infiltrated Fiber Mat). In

most applications fibers may act as secondary reinforcement used along with

conventional steel rebars or prestressing strands as main reinforcement. In the class of

high volume fraction of fiber, the materials have excellent mechanical properties and can

be used without other continuous reinforcement. However, these composites are often

suited for highly specialized applications due to limitations associated with processing

and cost.

1.2 Dissertation Objectives and Scope

The main objective of this study is to evaluate and accurately model the post-

cracking behavior of high performance fiber reinforced cementitious composites

(HPFRCCs). To achieve this objective, numerous sub-objectives are sought and include

identification uncovering the behavior of influential variables, namely: the stress and

strain at first cracking, the stress and strain at maximum post-cracking, the shape of the

stress-strain or stress-elongation response, the multiple cracking process, the shape of the

resistance curve after localization, the energy associated with the multiple cracking

process, the stress versus crack opening of a single crack, and the variability associated

with these parameters. This research will focus on both steel fibers and polymeric fibers,

which have the greatest potential with regard to current commercial application. The

objectives are accomplished through both an extensive experimental program and a

rational analytical program. Furthermore, a statistical regression analysis is performed to

provide information on the influence of important parameters. Figure 1.3 provides a

general flow chart of the research plan.

4

1.3 Research Significance

The significance of this research lies in the ability to predict with reasonable

accuracy the tensile stress-elongation response of fiber-reinforced cement composites and

their key properties, while understanding some of the mechanisms involved and the

variability associated with the resulting properties. The new information obtained will

lead to an improvement in modeling material properties for finite element analysis as

well as a greater confidence in using HPFRC composites in structural applications. The

overall research also provides a good foundation to integrate these composites in

conventional structural analysis and design.

1.4 Structure of the Dissertation

This dissertation is organized into nine chapters.

Chapter 2 provides a review of existing literature on the behavior of HPFRCC in

tension, namely: the first cracking point, the multiple cracking stage , the mechanism for

the maximum tensile strength, the localization stage, the impact of randomly distributed

fiber on tensile resistance, and crack opening and crack spacing when relevant.

Chapter 3 describes the experimental program for uncovering HPFRCCs behavior

under the direct tensile loading testing procedures followed, equipment setup used, and

parameters investigated.

Chapter 4 provides the details of the direct tensile test results using dogbone

shaped specimens. It includes discussions of the first cracking point, maximum stress

point, localization, crack spacing, and crack width. The direct tensile test results are also

analyzed in terms of their associated mechanisms.

Chapter 5 describes the tests related to the stress versus crack opening

displacement (σ-COD) using notched tensile prisms. Different fibers (PVA, Spectra,

Torex, and Hooked) and different volume fraction (0.5%, 0.75%, 1.0%, 1.5%, 2.0%,

SIFCON) are used. A model for the stress versus crack opening displacement relation

after localization is proposed. Extensive image analysis is carried out to shed light on

5

crack propagation and multiple cracking when present.

Chapter 6 describes the proposed model for the post-cracking behavior of

HPFRCC under tension and the influence of different types and volume fraction of fibers,

It covers first cracking stress, ultimate strength, and the computational method suggested

to predict the multiple cracking stage. Regression equations are given to predict total

crack length and crack spacing and width at crack saturation. Finally, the proposed post-

cracking model for direct tensile stress-strain relation is illustrated on a few examples.

Chapter 7 describes the statistical variation of results observed from the direct

tensile tests and the variability of each experimental variable. It also discusses how

variability affects the proposed tensile model.

Chapter 8 describes a newly proposed ring-tensile test, the results of a related

simulation based on a finite element analysis, and a parametric evaluation. The expected

advantages of this new test setup are discussed. Experimental results of a preliminary

investigation using the ring tensile test are provided and analyzed. Expansion or strain

measurements and a critique of the technique are described.

Chapter 9 presents a summary of overall research results and related conclusions.

A section on recommended future research is provided.

6

Figure 1.1: Stress-strain relation of typical tensile behavior of HPFRCC

Figure 1.2: Typical schematic tensile behavior of HPFRCC, i.e., strain hardening

behavior

7

Figure 1.3: Research plan and objectives

8

CHAPTER 2

LITERATURE REVIEW MODELING OF HPFRCC IN TENSION

2.1 Introduction

Figure 2.1: HPFRCC Stress-strain relation

9

Several attempts were made in the past to model the behavior of HPFRCC in

tension. Currently fiber reinforced material models can be classified into three categories,

composite mechanics approach, energy approach, and numerical approach. Each

approach represents HPFRCC behavior in a different way. Consequently, to model the

behavior of HPFRCCs, it is necessary to understand several important parameters. These

parameters relate in particular to the post-cracking response and the shape of the Stress-

Strain (or Stress-Displacement) relationship (see Fig. 2.1).

1. First cracking point (A) , (σcc, εcc)

2. Multiple cracking stage (A-B), (σcc, εcc to σpc, εpc)

3. Maximum tensile strength point (B), (σpc, εpc)

4. Localization and softening stage (B-C), (After σpc, εpc)

5. Crack-opening or crack width in HPFRCC

6. Crack spacing in HPFRCC

2.2 First cracking point (A), (σcc, εcc)

The first crack strength is defined as the applied tensile stress at which the crack

spreads throughout a cross-section. Naaman (1972, 1974, and 1987) introduced a

composite mechanics approach for modeling composites reinforced with short

discontinuous fibers. Each model treats the effect of discontinuity and randomness of

fiber orientation in a different way. Naaman (1974 and 1987) used the orientation effect

which was proposed by Romualdi and Mandel (1964).

In the model suggested by Naaman (1974), the fiber composite is modeled as a

chain link series which will break when the weakest link breaks. The fibers distribution in

the composite follows a Poisson distribution function. The cracking strength of one link

in the chain is given by the contribution of matrix and the contribution of fibers. The fiber

contribution depends on the fraction of bond mobilized at first crack and the coefficient

of fiber orientation in the uncracked state. The related equation (σcc = σmu (1-Vf) + α1 α2 τ

Vf L / d) is explained in Chapter 6.

10

Alwan and Naaman (1994) proposed two models for the elastic modulus of fiber

reinforced cement composites. The first model is based on the concept of interfacial bond

stress-slip, while the second model is a numerical scheme based on homogenization

theory. A good agreement between the analytical prediction and experiments was

observed.

Another model was proposed by Li and Lieung (1991) by using energy approach

and fracture mechanic. According to the principles of linear elastic fracture mechanics

(LEFM), stresses at the flaw tip are proportional to the stress intensity factor, KI, which

can be derived from basic principle of elasticity. In the model, the cracking stress was

derived based on fracture mechanics and micromechanics. The first crack state refers to

the first bend-over point and is assumed to be equal to the steady-state cracking stress σss.

In addition to this, the steady-state cracking stress is estimated from composite

crack bridging stress, which represents the apparent closing pressure due to fiber bridging

acting on the composite crack plane, and cracking stress level. Moreover, the cracking

stress level is defined as the stress level at which each of the multiple cracks propagates,

when each crack in a different part of the specimen has a different size.

2.3 Multiple Cracking Stage (A-B), (σcc, εcc to σpc, εpc)

For fiber reinforced composites, two possibilities exist after formation of the first

transverse crack: strain softening and localization characterized by continuous opening of

the major crack due to fiber pull-out (or fiber failure or both), and strain hardening

characterized by multiple cracking. For fiber reinforced cementitious composites,

multiple cracking is described as the stage when more cracks propagate along the

transverse direction parallel to the first transverse crack.

Generally, at the first cracking point (σcc, εcc), the specimen is assumed to have a

single crack. Consequently, the overall specimen stiffness will be reduced since the crack

is opened and elongated. Other parts of the specimen continue to be non-cracked

elements, representing the same stiffness as a typical elastic section. When the load is

increased, the crack will either open more, or new additional cracks develop. In the last

case, the numbers of cracks subsequently increases until the specimen reaches the

11

ultimate strength or saturation cracking.

For simplicity, it can be assumed that the multiple cracking stage (A-B) is a linear

relationship between the first cracking point (A) and the ultimate cracking point (B)

(Kanda, Lin and Li, 2002). However, the multiple cracking stage is not linear since when

the crack is formed, the stres versus crack-opening displacement (COD) is not linear and

the behavior also relates to several parameters including fracture toughness of the

composites, fiber pull-out behavior and fiber parameters (Yang and Fischer, 2005).

A major obstacle in tensile stress-strain modeling is the characterization of the

inelastic strain due to matrix cracking. This inelastic strain was originally investigated for

continuous aligned fiber-reinforced composites by Aveston et al. (1971), in which matrix

cracking stress was simply assumed uniform in each of the multiple cracks. Their results

were then extended for composites reinforced with randomly distributed long fibers

(Aveston and Kelly, 1973). Following these research studies, matrix cracking of

composites under tension has been extensively investigated in the field of ceramics.

Stress at cracking state was derived as a function of micromechanical parameters

representing the initial flaw size and the fiber’s crack bridging performance (Marshall et

al, 1985). Furthermore, cracking was statistically examined, and its stochastic aspects

were analytically clarified (Cho et al, 1992 Searing and Zok, 1993). Based on these

studies, inelastic strain due to matrix cracking was modeled in relation with crack

evolution. Also an analytical model for the stress-strain relation was then proposed for

ceramic composites, which applies to the case where fibers are aligned and continuous.

However, few attempts have yet been made to extend these theories for fiber-reinforced

composites with randomly oriented short fibers.

Tjiptobroto and Hansen (1993) proposed a tensile strain hardening and multiple

cracking models for HPFRCC based on the ACK theory (Continuous fiber reinforced

composites theory), energy concept and the shear-lag model. In their research, the

occurrence of multiple cracking in discontinuous fiber reinforced composite depends on

the energy. If the energy required to open an existing microcrack is larger than the energy

required to form a new micro crack, then a new crack will occurred. It is assumed that the

magnitude of the energy for every multiple crack is the same. The strain at the end of

multiple cracking represents the sum of both the elastic strain and the inelastic strain. The

12

researchers admitted that, in reality, it is expected that the energy required to create the

first crack is smaller than the energy required to create the following cracks due to the

random strength distribution of the matrix.

Kanda, Lin and Li (2000) proposed a tensile stress-strain model with emphasis on

fundamental micromechanics and experimental observations of multiple cracking. The

theory represents an extension of the approach adopted by Wu and Li (1995), in which

the multiple cracking sequences was treated statistically and the flaw size distribution

was simulated by a Monte Carlo technique (Wu and Li, 1995). However, their model

assumed the multiple cracking stages (σcc, εcc to σpc, εpc) to follow a simple linear

relationship between the first cracking state and the ultimate tensile state.

Akkaya, Shah and Ankenman (2001) studied HPFRCC characterization on PVA

fiber composites. The study investigated the effect of fiber dispersion on the sequential

multiple cracking of fiber-reinforced cement composites. The electronic speckle pattern

interferometry (SPI) technique was used to observe the multiple cracking of the material

and to evaluate the cracking stresses of the composite. They reported the success of the

technique to capture the highly accurate displacement measurement, which allowed

mapping of the crack propagation at the microscale, determining the number of cracks

and the location of the cracks precisely. They also concluded that the fiber dispersion

affected the toughness of the composite. An effective crack bridging and increase in the

toughness of the composite can be achieved if the fiber dispersion is better at the first

crack location. However, no multiple cracking prediction was suggested in their research.

Yang and Fischer (2005) investigated the fiber bridging stress-crack opening

relationship of fiber reinforced cementitious composites. The experimentally obtained

information on the fiber bridging stress-crack opening relationship is used to assess the

potential for multiple cracking and strain hardening behavior of HPFRCC in uniaxial

tension. They proposed a method to simulate the displacement-controlled uniaxial tensile

behavior of HPFRCC using information from its stress-crack opening relationship. The

multiple cracking phenomenons in the strain hardening stage is captured in the simulation

results and characteristics of multiple cracking such as crack width and spacing can be

obtained. The preliminary simulation results have shown promising results. However, the

researchers admitted that there is no verification between the simulation results and

13

experimental observation.

2.4 Maximum Post-Cracking Tensile Strength Point (B), (σpc, εpc)

The maximum post-cracking tensile strength point (σpc, εpc) is the point at which

maximum stress occurs prior to localization failure and softening of the response.

Generally, it is the state where the multiple cracking behavior stops (no occurrence of a

new crack) and the localization failure starts (the critical crack start opening while other

cracks unload).

Several attempts were made in the past to predict σpc and εpc. Naaman (1972,

1974, and 1987) proposed a prediction equation for the post-cracking strength as a

function of shear strength, volume fraction of fibers, fiber length and diameter of fiber.

Later Naaman expanded the post-cracking strength so the model can be generalized and

applied to high performance fiber reinforced cementitious composites for which the post

cracking strength can be greater in value than the first cracking strength; these include the

modification factor for the expected pull out length, the efficiency factor of orientation in

the cracked state, the group reduction factor associated with the number of fibers pulling-

out per unit area, the pulley effect, the factor for fiber at large angles, and matrix

damage. In short, the maximum post-cracking strength can be expressed as

dLV fpc λτσ = . (λ = λ1 λ2 λ3). (This equation is explained in Chapter 6)

Li and Lieung (1992) proposed a model to predict maximum stress (σpeak) as a

function of volume fraction of fiber, bond strength of fiber and matrix, the length of fiber

and also the diameter of fibers. Also they introduced the probability factor accounting for

fiber distribution and the snubbing effect in their model. However, their model has no

group effect of fibers, the interaction that occurs between two or more fibers, reducing

the performance of each individual fiber to resist the tensile load. Consequently, their

model seems to largely overestimate test results.

Later the model proposed by Kanda, Lin and Li (2000) to predict the strain at

maximum stress (εpc) in which the stress-strain relation is assumed to be a bilinear. This

approach requires theoretical modeling of the ultimate crack spacing theorydx . The specific

14

case of HPFRCC involving full saturation of multiple cracking has been studied (Kanda

and Li (1998)). The ultimate crack spacing with full crack saturation xd (saturated

ultimate crack spacing) depends on the transferred stress from bridging fibers at the crack

plane to the non cracked matrix. This stress transfer is achieved via the interfacial shear

stress between fibers and matrix. The saturated ultimate crack spacing xd can be estimated

as the minimum distance necessary for overcoming matrix cracking stress by transferred

stress. The theory was conducted under the hypothesis that the initial flaw size

distribution can be represented as a random process that governs crack evolution.

According to the crack spacing theory, it is assumed that unsaturated crack

spacing can be evaluated by employing the probability of a potentially propagating flaw

involved in a tensile specimen. Cracks are not to be generated with spacing less than xd,

as in the theory, the matrix stress cannot attain cracking stress level within length xd from

a crack plane.

The model employs a probabilistic description, in which the Weibull function is

adopted to represent flaw size distribution. Identifying the parameters in this function

requires at least one set of experimental data on crack spacing. In this study, the

researcher claimed that their study revealed that the flaw size distribution can be assumed

unique for similar composites (with identical matrix mix proportion, mixing process and

age) with different fiber length and fiber volume fraction and the ultimate crack spacing

is found to be consistent with the results predicted by means of the proposed crack

spacing theory

The proposed model can achieve the first step to solve the problem of ultimate

crack spacing ( theorydx ) and the strain at ultimate stress. However, based on many

experimental results of Spectra fiber, the accuracy of this estimation is not completely

satisfactory. Their model seems to be overestimated in both stress and strain at

maximum stress (σpc, εpc); the error is the result of complexity of formula accounting for

both fracture mechanics and energy method, which required numerous parameters

including matrix fracture toughness, slip-hardening parameters in pull-out test. Also the

model did not include the group effect of fiber, the interference of numerous fibers

confined in small area, which may explain the overestimation of results.

15

Another model was developed by Tjibtobroto and Hensen (1991,1993), which is

based on extending the energy-based ACK model (Aveston et al, 1971) of continuous

fibers and the shear-lag model of single fiber pull-out process to fit cement composites

reinforced with short discontinuous fibers. By comparing the energy required to form a

new crack with the energy required to open the first crack, the model can represent the

strain at maximum stress (εpc) as a function of elastic modulus of fiber, elastic modulus of

composite in multiple cracking stage, maximum elastic strain of the composite, or its first

cracking strain, debonding energy and constant frictional interfacial bond stress.

However, the verification of this model seems to be overestimated. The error may come

from the finding the elastic modulus of the composite in multiple cracking stage, which

can be obtained in several ways and is not constant during the multiple cracking stage.

Moreover, Tjibtobroto and Hensen (1991, 1993) verified their model by using

beam-flexural tests instead of direct tensile tests. The verification was performed with

very high volume fraction fiber-reinforced specimens (FR-DSP, Vf = 4% - 16%) with a

very fine matrix. There is no verification with low volume fraction fiber-reinforced

composites. Also the verification was conducted only on the flexural tests with small

strain gauges attached to the top and bottom part of the beam. Unfortunately, no

verification was carried out using a direct tensile test.

Kullaa et al (1998) proposed a model for maximum stress and strain at maximum

stress (σpc, εpc) based on several assumptions such as matrix cracks are planar and

perpendicular to the load, the fibers are separate, straight, smooth and fully flexible in

bending, the frictional shear stress is constant or decaying along the debonded length and

both the fibers and the bulk matrix behave in linear elastic fashion up to their tensile

strength. The maximum stress model is similar to the model proposed by Li et al (2000).

The strain model is an extension of the ACK theory by Aveston, Cooper and Kelly as it

can be used with discontinuous fibers with different distribution. However, the prediction

of strain at maximum stress (εpc) is the combination of elastic strain and plastic strain

where the plastic strain depends on matrix cracking which requires a parameter

accounting for the matrix spalling length. The researcher admitted that the spalled length

is still to be determined and the model is not fully developed. However, based on the

researcher verification, they admitted that the predicted maximum stress and strain at

16

maximum stress are relatively smaller than experimental observations.

2.5 Localization Stage (B-C), (After σpc, εpc)

Normally, the localization stage of the HPFRCC occurs after the peak (σpc, εpc).

At this stage, the tensile resistance significantly drops with increased displacement or

crack opening. No new cracking occurs, and the critical crack opens while other multiple

cracks are closing since the stress decreases causing strain-softening. The characteristics

of this stage depend on the pull-out behavior of the composite. In this process two

constitutive relations can be distinguished, the stress-strain curve and the stress-crack

opening curve. The former consists of the normalized sum of crack widths equal to all of

the closing cracks as well as the strain in the matrix. The latter consists of the critical

crack width of the localization crack. In analysis, these two constitutive relations should

be distinguished.

A comprehensive model for crack-opening behavior should cover the full possible

range between complete fiber pull-out and complete fiber rupture. The lower bound

model should come from the assumption that all fibers fail (no ductile behavior). The

upper bound model should come from the assumption that the fibers, virtually all aligned

in one direction, pull out perfectly and the crack opens perpendicular to the fiber

direction. It is also assumed that a certain proportion of the crack width is permanent. The

width of the opening crack is increased gradually while the others are closing. However,

complete crack closure cannot occur upon unloading due to a reversed frictional stress

which prevents fibers from slipping back into the matrix when the specimen are unloaded

(Wu et al., 1994)

The anticipated model for the localization stage was founded upon Visalvanich

and Naaman (1983) by proposing the post-cracking strength model. The f(δ) function is a

stress versus crack-opening function. By modifying the maximum post-cracking stress

and half the fiber length l/2 as the normalizing factors, respectively for the stress and

displacement axis, an analytical relationship representing the stress-crack opening law of

crack-opening composites was derived and representing as a behavior of tensile

responded after maximum stress.

17

In the model of stress versus crack-opening for tensile fracture in SFRC

conducted by Gopalaratnam and Shah (1987), an experimental program on notched

specimens of SFRC was performed. They concluded that after catastrophic debonding, a

constant frictional stress transfer ensures adequate post-cracking strength. Thereafter,

fiber slip resulting from the widening of the single matrix transverse crack yields a

composite stress-displacement relationship that is essentially linear with a negative slope.

Also the composite stresses can be obtained by superimposing solutions for individual

fiber pull-out with the solution for matrix softening after accounting for the randomness

in fiber length and orientation, and the compatibility of displacements at the crack.

Consequently, the model developed from experimental results and analytical model of

single fiber pullout seems to be realistically sensitive to important reinforcing parameters.

However, the researcher admitted that the effects of the plastic bending of fibers and of

the matrix crushing due to pull-out of inclined fibers were not considered.

2.6 Crack-Opening in HPFRCC

In addition to the post-cracking mechanism, the crack-opening displacement

model (COD) is vital to determine the overall strain behavior of HPFRCC in the multiple

cracking stage (A-B). The average strain in the composite is directly related to the crack-

opening displacement at every step of crack propagation.

Generally, for fiber reinforced cement based composites, the fiber bridging force

would not be constant but would depend on the opening and extent of the crack. It is

well accepted that crack propagation, in cementitious materials and in their composites, is

controlled by the formation of a crack-bridging zone behind the crack tip. This bridging

zone is often referred to as the fracture-process zone, in which microcracking and

inelastic deformations have occurred. Because of its dominant role, it is crucially

important to determine the law that governs the formation of the fracture-process zone

(i.e., the bridging-stress-crack-opening relationship).

Normally, the crack comprises three different zones. (Fig. 2.2)

18

Zone 1 represents the real crack in the composite along which the cracked

surfaces are under zero resisting stress because all the bridging fibers have either pulled

out or failed.

Zone II is defined as the pseudo plastic zone. It is a zone where the matrix has

cracked, but the bridging fiber still provides some resistance to pullout or tensile

stretching.

Zone III is a microcracking zone or a process zone of the composite. It is a zone

where elastic and inelastic deformations occur.

Naaman (1972) studied a statistical theory of strength for fiber reinforced

concrete using dogbone-shaped tensile specimen (i.e., with notch in the middle to control

the location of the crack position, which occurred during the test) and double cantilever

beams. He concluded that the setup can be used to measure some fracture properties of

fiber reinforced concrete and primarily estimate the size of a pseudo plastic zone

corresponding to an area where the matrix is cracked. In addition, he concluded that the

fibers are pulling out and the maximum crack tip opening is equal on average to half the

fiber length.

Later Visalvanich and Naaman (1983) extended the model for fiber reinforced

concrete by carrying out an extensive investigation dealing with modeling fracture in

fiber reinforced cementitious composites. They found that there is strong evidence that a

crack in a cementitious composite, subjected to a tensile stress field, starts propagating

when the crack-opening angle reaches a critical value, regardless of the crack length and

crack shape. The advancing of crack in a cementitious composite can be assumed to

maintain the same shape, provided the specimen size is sufficiently large and the crack is

not blunted. Also, they concluded that the post-cracking stress of steel fiber reinforced

mortar (with pullout behavior) gradually decreases from a maximum value to zero, while

the corresponding displacement or crack opening increases from that of the composite at

cracking to about half the fiber length. The post-cracking energy of the material is

substantially larger than its pre-cracking energy, which will control the fracture behavior

of the material.

Wecharatana and Shah (1983) studied the fracture resistance of fiber reinforced

concrete by using both double torsion specimens and notched beam specimens. The

19

classical linear elastic fracture mechanics analysis was modified to predict fracture

energy (R-Curves) for fiber-reinforced cement based composites. The predicted results

compared well with the experimental data from both types of tests. They introduced the

crack closing pressure model from the fracture energy and experimental observation.

They also concluded that the model is corresponding to the pull-out resistance of fibers.

However, their experimental observation and analytical model did not represent the

crack-opening displacement relation before the peak stress, only the stress after cracking

of the matrix.

Gopalaratnam and Shah (1987) reported the tensile failure of steel reinforced

mortar from notch testing (COD) to study the crack width of SFRC reinforced with

Hooked 0.5%, 1.0%, and 1.5% volume fractions of fiber. In their study, the crack

opening displacement was captured and compared with analytical predictions. They

conclude that the softening behavior of SFRC concrete, mortar, and paste seem primarily

related to the widening of a single crack. Measurements of optical crack widths, local

strains, and residual deformations all confirm the observation. An analytical model

proposed for both the fiber pull-out problem and tensile fracture of SFRC accounting for

all the major phenomenological processes of failure in such composites was suggested.

However, they concluded that the model is realistically sensitive to important reinforcing

parameters but should be improved for further study.

Li et al (1991) suggested that the composite bridging stress-COD curve can be

approximated by summing the contributions of the individual fibers bridging the matrix-

crack plane, by using the probability-density function of the orientation angle and

centroidal distance of fibers from the matrix plane.

Assuming a purely frictional fiber/matrix interface and complete fiber pullout,

Visalvanich and Naaman (1983) derived a semi-empirical model for the tension-softening

curve in discontinuous randomly distributed steel-fiber-reinforced mortar. With the same

assumptions, Li (1992) derived an analytical model taking into account an additional

frictional effect called the snubbing effect. The model provided a good prediction for the

post-cracking strength, the tension-softening curve, and composite fracture energy for a

number of composites in which the fibers did not rupture. However, discrepancies were

observed between the prediction of this model and some experimental measurements,

20

which suggest the occurrence of fiber rupture.

Maalej, Li, and Hashida (1994) developed a new technique to determine the

fracture process zone based on the J-integral analysis. The σ-δ relationships deduced

from the J-based technique have been compared with results of uniaxial-tensile tests.

Later Maalej, Li, and Hashida (1995) studied the effect of fiber on tensile

properties of short fiber composites by using notch-tensile specimens. In their studies, a

micromechanical model for the composite bridging stress-COD relationship that accounts

for fiber pull-out and tensile rupture was presented. The model accounts for a local

frictional effect and snubbing; however, the model does not account for fiber-bending

rupture and the possible effect of matrix spalling at the exit point of inclined fibers from

the matrix. The model assumes a fiber/matrix interface that is controlled by a constant

frictional bond stress. The post-peak curve predicted model is in good agreement with

the experimental result. However, the researcher admitted that if the fibers are brittle,

bending rupture will also need to be included in a composite model.

Nelson, Li, and Kamada (2002) studied the fracture toughness of Microfiber

reinforced cement composites. They attempted to quantify the reinforcing ability of

polypropylene (PP), polyvinylalcohol (PVA), and refined cellulose (RC) fibers in the

frontal process zone. They reported the successful experimental observation of the crack

by using an optical microscope, which was employed to visually monitor crack formation

during fracture toughness testing. The 50X magnification optical microwatcher lens was

focused on the notch tip. This allowed the visually observed frontal crack processes to be

correlated directly with the nominal tensile stress, and the crack mouth opening

displacement.

Yang and Fisher (2005) studied the fiber bridging stress-crack opening

relationship of fiber reinforced cementitious composites by investigating Engineered

Cementitious Composites (ECC) with PVA fibers (2%) by using double notch specimens.

They observed that the stress-crack opening curve becomes more consistent with

increasing the curing time of the specimen. The experimental results indicated a

relatively small variation in peak bridging strength over time. Furthermore, the

comparison of response at different curing times shows that the displacement, as well as

the total energy absorbed in the fiber bridging-crack opening process, decreases

21

significantly when curing time increases. They successfully applied the stress-crack

opening curve to simulate the multiple cracking and strain hardening behavior for ECC

under direct tension.

2.7 Crack Spacing in HPFRCC

The crack spacing in HPFRCC composite is the distance between two consecutive

cracks. Also the average crack spacing at saturation of cracking is a key parameter to

determine the equivalent strain at maximum post-cracking stress (εpc). From

experimental observation, the average crack spacing at the beginning of the multiple

cracking processes are relatively high. However, after the specimen elongation increases,

the number of cracks that occur increases. Consequently, the average crack spacing in

HPFRCC decreases until the composite reaches a saturation state generally at or before

the maximum stress (σpc). After this point, localization failure will occur and the average

crack spacing (excluding the localization crack) remains constant.

Yan, Wu and Zhang (2002), studied the cracking pattern in SIFCON. In their

research, the SIFCON volume fraction of fibers ranged from 4 % to 10 %. Digital image

analysis was conducted to capture the cracking pattern. They concluded that the higher

the volume fraction of fibers, the larger the amount of crack and the maller the crack

spacing.

Aveston et al. (1971) first proposed the conditions for multiple cracking in

continuous aligned fiber reinforced brittle matrix composites, notably known as the ACK

theory, and lay the foundation for subsequent research (Li et al, 1991-1992, Marshall et

al., 1985). In these models, uniform distribution of identical flaw size in the matrix is

implicitly assumed. Consequently, a deterministic composite strength during multiple

cracking is predicted, and usually does not agree well with experimental findings, as

observed for discontinuous random fiber reinforced cementitious composite (Wu and Li,

1995). This discrepancy is also found in a continuous aligned SiC reinforced calcium

aluminosilicate (Cho et al., 1992; Yang and Knowles, 1993). A rising load carrying

capacity beyond the first cracking strength during multiple cracking resembles strain

hardening of metals, often referred to as pseudo strain-hardening. Multiple cracks

22

develop over a wide range of load levels. Besides the difference in observed stress-strain

curves, the distribution of crack spacing also shows evidence against the assumption of

identical flaw size. A simple calculation was used to compute the crack spacing, x, on the

basis of force balance (Aveston et al., 1971; Wu and Li, 1992).

2

22 XLLLx ffftheory

d

πψ−−= (2.1) (Li et al, 1992)

Where

gπ

ψ 4= (2.2),

if

fmum

VdV

xτ

σ4

= (2.3)

where Lf is the fiber length, ω (=4/(πg)) is the correction factor for 3-D fiber

randomness, g is the snubbing factor.

Hence, after crack saturation, final crack spacing between x and 2x is expected.

However, from the comparison when applying to the SIFCON that was conducted

(section 3.9 step 2), largely underestimation is observed.

Kimber and Keer (1982) have calculated the average value of the crack spacing

for multiple matrix cracking when the matrix strength is a deterministic value, based on

analogy with minimum average spacing between cars of length x parked at random in an

infinite line. Their results yield an average spacing of 1.337x. However, experimental

observations of multiple crack spacing in steel/epoxy (Cooper and Sillwood, 1972), in

carbon/glass (Yang and Knowles, 1992), and in SiC/calcium alminosilicate (Cho et al.,

1992) do not support this prediction. Instead, better agreements between theory and

experiment are found when a statistical distribution of matrix cracking strength is

assumed (Cho et al., 1992; Cooper and Sillwood, 1972; Yang and Knowles, 1992).

In the ACK model, the energy balance at onset of cracking was used to predict the

elastic strain capacity of the composite reinforced with continuous fibers. Tjiptobroto and

Hansen (1991) modified the crack spacing model for discontinuous short fiber

23

composites by using the energy concept similarly to the model used in the ACK model.

The changes in the external work were assumed to be equal to the changes in internal

energy. The energy changes are obtained by evaluating the different energy terms before

and after the occurrence of a crack. The major difference between the ACK model and

Tjiptobroto and Hansen (1991) model for discontinuous fibers is in the combination of

the different energy terms associated with internal energy changes due to the occurrence

of a crack. This method will also be modified in this research to determine the ultimate

crack spacing from energy concept.

Regarding the nature of the distribution of crack spacing in composites, Wu and

Li (1992) proposed the hypothesis that the strength distribution may comes from a

distribution of matrix flaw sizes, interaction of matrix cracks and variation of fiber

reinforcement.

Later Wu and Li (1995) studied the stochastic process of multiple cracking in

fiber composites. In their research, the multiple cracking in discontinuous random fiber

reinforced brittle matrix composite is studied and simulated by a Monte Carlo simulation.

A good agreement of composite strength and average crack spacing between

experimental data and simulation results is found for a polyethylene fiber reinforced

cement paste (Vf = 2%). They suggested that the condition of crack propagation depend

on initial flaw size, external load, and fiber bridging effect. However, they found

differences between their prediction and experimental results and concluded that they

may come from the effects of crack interaction, which were not considered in their study.

Kanda, Lin and Li (2000) again proposed a model for crack spacing by modifying

the ACK theory (Aveston et al., 1971; Wu and Li, 1992) and using a statistical

distribution function to account for flaw size distribution, crack interaction, crack

characteristic and fiber volume fraction. However, many parameters are required from

experimental observations and need to be known to calibrate the model. They also

suggested that the actual crack spacing should be higher than the crack spacing obtained

from the modified ACK theory. In conclusion, they suggested that it is possible to predict

the crack spacing reasonably accurately.

In this research, the equivalent number of cracks in a typical specimen, the

average crack spacing and width at saturation cracking are addressed in Chapter 6.

24

Figure 2.2: Crack model for HPFRCCs showing (a) fracture zone and (b) possible stress

distribution (from Visalvanich and Naaman, 1983)

25

CHAPTER 3

EXPERIMENTAL PROGRAM FOR EVALUATING HPFRCC TENSILE BEHAVIOR

3.1 Introduction

As described in Chapter 2, considerable research has already been carried out to

evaluate the response of HPFRCCs in tension (Swamy and Mangat, 1974; Visalvanich

and Naaman, 1983; Naaman et al, 1987, 1992, 1996; Nammur, 1989; Li and Lieung

1991; Mishra 1995; Lin and Li, 1997; Kullaa et al, 1998; Kanda, Lin and Li, 2000;

Kabele, 2004; Tjiptobroto and Hansen, 1991, 1993; Alwan and Naaman, 1994;

Chandrangsu and Naaman, 2003). Based on these and other results, micro-scale models,

which involve a direct description of three phases of materials, i.e. fibers, matrix and

interfacial zone, have been introduced. However, an entire model for HPFRCCs behavior

in tension is not yet available (e.g., the linear-elastic stage, multiple cracking stage, and

the localization failure stage) and none of the existing models have the capability to

represent the effect of statistical variability and the accuracy of the predicted behavior.

Experimental studies (direct tensile test, crack-opening displacement test, ring

tensile test) were conducted to clarify the parameters for each stage of the response and

lead to the development of a complete model for HPFRCs. Data from the experimental

programs were processed in analytical works, and were utilized in the proposed model.

26

Figure 3.1: Flow chart of experimental program

3.2 Materials

3.2.1 Mortar

The cementitious composite used throughout this study is a regular mortar, having

an unconfined compressive strength of about 56 MPa (8.1 ksi). Cement utilized

throughout this research was commercially available Type III Portland cement

manufactured by Holcim Cement Company, which provides a rapid developing strength.

Evaluation of post-cracking tensile behavior of HPFRCCs

(σcc , εcc)

(σpc , εpc)

Multiple cracking

Direct tension

Ring tension

Implementation of prediction model and verification

Data collecting and characteristic Analysis

HPFRCC stress-strain relation

COD notch test

27

However, this type of cement generates an elevated hydration heat compared to other

types of Portland cement. Consequently, the stresses in the concrete at an early age could

be significant. A summary of the cement compositions and properties provided by the

manufacturer is presented in Table 3.1.

Table 3.1: Cement compositions and properties of Portland cement type III

The mixture ratio, based on weight of cement, sand, fly ash, and water is 1, 1, 0.15,

and 0.35, respectively. In addition to this, some admixtures, such as VMA, air entrapping

agent, and superplastizer, were applied to improve the behavior (mixing, compacting) of

HPFRCC mixtures.

Table 3.2: Mix proportion of matrix by weight

3.2.2 Fibers

Four types of fibers were investigated in this study, two polymeric fibers and two

steel fibers. Their properties cover a wide range of mechanical properties, as illustrated

in Table 3.3. A detailed description of each type of fiber is described in the following

section.

28

Figure 3.2: Fibers used in present research, (a) PVA, (b) Spectra, (c) Hooked, and (d)

Torex

Figure 3.3: Fibers used in present research: length comparison

(a) PVA (b) Spectra

(c) Hooked (d) Torex

29

Table 3.3: Fiber properties

3.2.2.1 Polyvinyl Alcohol (PVA)

Polyvinyl Alcohol (PVA) fibers have been used on a large scale in the

construction of thin products as a replacement for Asbestos fibers due to their good

mechanical properties and physical characteristics.

A monofilament PVA fiber was used in this study, possessing a tensile strength

about half to one third of that of Spectra fiber. Furthermore, the monofilament PVA fiber

costs approximately 1/8 of Spectra fiber. Generally, the fiber section is close to circular.

Its diameter was 0.19 mm and its length was 12 mm. High chemical bond strength, due to

the hydrophilic nature of PVA fiber, is a property characteristic of PVA. This high

chemical bond strength is due to a strong hydrogen intermolecular bond resulting in high

bond strength between PVA fiber and cementitious materials. However, the high bond

strength and relatively low tensile strength lead to a fiber rupture tendency during the

opening of a matrix crack, rather than a more desirable pull-out process. Therefore, an

oiling agent is usually applied onto the fiber surface during the production process to

reduce bond (Li et al., 2002). However, the long term effectiveness of such oiling agent

has not been demonstrated.

30

Figure 3.4: PVA fiber

Figure 3.5: PVA fiber in composites

3.2.2.2 Spectra

Spectra fiber (Trade name) is made from ultra-high molecular weight

polyethylene. It has outstanding strength and toughness, but a relatively weak bond

strength. Spectra fiber material is referred to as ultra high modulus PE (UHMPE), which

is produced with a very high molecular orientation by gel spinning, and subsequent

drawing which yields fibers having up to 85% crystalline structure with 95% parallel

orientation. The polymer chains of UHMPE are bound together at various points by

mechanical cross linking. This produces strong inter-chain forces in the resulting

filaments that can significantly increase the tensile strength. The filaments emerge with

an unusually high degree of orientation relative to each other, further enhancing strength.

Although the bond strength of Spectra fiber is relatively weak, it exhibits a slip-hardening

behavior during fiber pullout, which is caused in part by an abrasion effect. When a

Spectra fiber is pulled out from a cementitious matrix, the fiber surface is damaged and

stripped into small fibrils due to the abrasion effect. These small fibrils jam the tunnel

surrounding the fiber, which in turn prevents the fiber from being pulled out. This

mechanism significantly increases the frictional bond strength between the Spectra fiber

and cement matrix, which leads to a slip-hardening response. In conclusion, having a

tensile strength as high as steel fiber, but seven times lighter in weight, Spectra fiber is

one of the toughest and lightest polymetric fibers. In this study, a Spectra fiber having a

length of 38 mm and a diameter 0.038 mm diameter was investigated.

31

Figure 3.6: Spectra fiber

Figure 3.7: Spectra fiber in composite

3.2.2.3 Hooked

An industrial Hooked fiber, commercially named “Dramix” from Berkaert S. A.,

Belgium, was used in this study. The Hooked fibers are made from cold-drawn, high

tensile strength steel wire and have Hooked ends. Fibers with two different tensile

strengths were investigated. One Hooked fiber had circular cross sections with diameter

of 0.3 mm, length of 30 mm, tensile strength of 1050 MPa, and Hooked ends (regular

strength Hooked fiber). The other fiber also had circular cross section with diameter of

0.4 mm, length of 30 mm, tensile strength of 2100 MPa, and Hooked ends (regular

strength Hooked fiber). In this regard, steel fibers have the advantage over other fibers in

terms of ease to be deformed to improve their mechanical bond, as well as high tensile

strength and ductility. The mechanical bond of steel Hooked fibers derives from the

Hooked ends, which contribute to bond strength through the work needed to straighten

the fiber during pull out. A detail of the mechanisms of how Hook ends behave during

pull-out is presented in Figs. 3.10-3.11

32

Figure 3.8: Hooked steel fiber (before

mixing)

Figure 3.9: Hooked steel fiber in

composites

Figure 3.10: Pull-out mechanism in

Hooked steel fiber

Figure 3.11: Formation of two plastic

hinges for maximum resistance

33

3.2.2.4 Twisted Polygonal Steel Fibers (Torex)

A newly developed steel fiber of optimized geometry called “Torex” (Twisted

Polygonal Steel Fibers) was introduced by Naaman at the University of Michigan with a

subsequent U.S. patent No. 5989713 (1999).

Figure 3.12: Torex fiber geometry

A prototype machine developed especially for research was used to produce twisted

polygonal steel fibers for testing HPFRC composites. To make a twisted fiber, this

machine takes a round wire, shapes it, twists it and then cuts it in a continuous process.

The speed of each component of the machine can be controlled: thus, the fibers can be

made with different numbers of twists and lengths. Moreover, studies by Naaman and

Guerrero (1999), Naaman and Sujivorakul (2002), Kim and Naaman led to development

optimized geometry for various material parameters; moreover, Torex fibers offer a ratio

of lateral surface area to cross sectional area larger than that of round fibers. Torex fiber

is made of high strength steel wire with a polygonal cross section, and twisted along its

length. The fibers are characterized by the shape of their cross sections and the number of

twists or ribs per unit length.

The larger surface area of Torex fibers leads to a direct increase in the

contributions of the adhesive and frictional components of bond. Moreover, unlike a fiber

with a round section, a fiber of polygonal section can be twisted along its longitudinal

axis, developing ribs, creating a significant mechanical bond component. The outstanding

advantage of Torex fiber is in its pull out resistance, which increases with an increase in

34

slip while being pulled out from the matrix. Torex fibers can maintain a high level of

resistance up to slips representing 70% to 90% of embedded length. This unique bond-

slip behavior is due to the successive untwisting and locking of the fiber embedded

portion in the tunnel of the matrix during slip. Photos of Torex fibers used in this study

are shown in Figs. 3.13-3.14.

Figure 3.13: Torex steel fiber

Figure 3.14: Torex steel fiber in

composites

Two types of Torex fiber were utilized in this study, regular strength Torex fiber

(tensile strength of 1380 MPa), and high strength Torex fiber (tensile strength of 2760

MPa).

3.3 Specimen Preparation

All specimens of this research were prepared in Plexiglas molds. During the

mixing, care was taken to prevent clumping of the fibers. The molds were lightly oiled

before pouring of the HPFRCC materials. The dry components of the mortar mix were

first combined with approximately 25% of the total water required. The fibers, along with

the remaining 75% of the water, were intermittently added as the mixing process

progressed. The fibers were added slowly, while mixing continued, in a sprinkling

fashion in order to distribute the fibers thoroughly throughout the mix. Particular care

was paid while adding the Spectra fibers because it was observed that Spectra fibers

could potentially trap large amounts of air in comparison with the other fibers used in this

study. After casting, the specimens were kept under preventive cover while remaining in

35

the molds for 24 hours. After that, the specimens were removed from the molds and

placed to cure in a water tank for at least 14 days. Next, they were left to air dry for a

period of at least 48 hours prior to testing. After drying, thin poly-urethane spray was

applied to the surface of the specimens to aid in crack detection during and after testing.

It should be noted that HPFRCC reinforced PVA specimens did not use exactly the same

mixture as referred to in some ECC specification, (Engineered Cementitious

Composites), in order to provide a fair comparison with other fibers.

3.4 Direct Tensile Test (Dogbone Test)

To obtain increased understanding about HPFRCCs behavior program, under

monotonic direct-tensile load, a conventional dogbone shape specimen (Figs. 3.15, 3.16)

was selected to detect elongation occurring during the test. Two Linear Variable

Differential Transformers (LVDTs) were attached near the grips connected to the

specimen (Fig. 3.17). The top and bottom ends of the specimens were held by specially

designed grips attached to the MTS 810 machine. The average elongation was obtained

from the two LVDTs placed on opposite sides of the specimen at a predetermined gage

spacing (about 178 mm or 7.0 inches). The strain was calculated from the gauge length.

The tensile stress σt and tensile strain εt were calculated from the following equations:

AT

t =σ (3.1)

g

gt L

LΔ=ε (3.2)

where T is the tensile load obtained from the load cell, A is the cross sectional

area of the specimens, gLΔ is the elongation of the tensile specimen (average value from

the two LVDTs), and gL is the gauge length of the tensile specimens (178 mm or 7.0

inches).

Additionally in some series of tests, Optotrak sensors were placed on the

specimens’ surfaces to also measure deformation precisely at the surface (and ascertain

the values obtained from the LVDTs). Moreover, some specimens were monitored via

digital camera to observe closely the specimen cracking behavior. Furthermore, the

36

cracking patterns, crack spacing, and crack width were investigated.

Figure 3.15: Specimen dimension of tensile dogbone specimens (mm)

Figure 3.16: Specimen dimension of tensile dogbone specimens (inch)

37

(a) (b)

Figure 3.17: Tensile test setup, (a) Specimen ready for testing and (b) Instrumentation

Figure 3.18: Dogbone specimen

Figure 3.19: Mold for dogbone specimen and

added end reinforcement

38

A servo controlled hydraulic testing machine (MTS 810) was was used for all the

direct tensile tests as well as all the tests using notched prisms.

A deformation controlled procedure was used to capture the tensile response at a

rate of 5 data points per second. A deformation rate of 0.05 inch per minute (1.27 mm per

minute) was applied in the test.

Usually, the testing time for one specimen was between 2 to 10 minutes

depending on the experimental type of fiber, ductility, and energy absorption. If the

specimen was brittle, the time for testing was short. However, if the specimen was very

ductile, testing time lengthens. Normally, if the specimen has not completely failed the

elongation was increased up to about 0.3 inch (7.62 mm) prior to stopping the test.

3.5 Crack Opening Displacement Test (COD, Notched Prism Test)

To study the behavior of HPFRCC in tension, it is necessary to study the stress

versus crack opening displacement of the composite, crack width, crack opening, and

crack propagation in a specific location. The specimen in this type of test should have a

predetermined location of a crack. Thus a tensile prism, similar to the dogbone tensile

specimen, was used with a double notch at its midlength. The notched section had the

same dimensions as the gauge section of the dogbone tensile prism (Figs. 3.20 and 3.21).

Figure 3.20: Dimensions of notched tensile prim for crack-opening displacement test

(mm)

39

Figure 3.21: Dimensions of notched tensile prism for crack-opening displacement test

(inch)

Notched prism’ dimensions were (76.4 mm), 3 inches wide, 304.8 mm, 12 inches

long, and 25.4mm, 1 inch thick. Furthermore, the specimens had two notches, 12.7mm,

0.5 inch deep and 2.54mm, 0.1 inch wide, at the left side and the right side of the

specimen. The notches were cut using a circular diamond concrete saw. Microscopic

observations at 50X were carried out to verify that no micro-cracking was introduced in

front of the notches, due to the cutting process. Consequently, the crack occurred during

the test somewhere in between the two notches in the middle of the specimen, which is

the weakest location of the specimen.

Two crack gauges were placed at the left side, and the right side of the notched

specimen to record the displacement, which occurred at the notches. A camera was used

to closely observe the crack propagation and cracking behavior at the crack, and the area

in close proximity to the crack. The displacement and load data were linked to the camera

system, load cell, and data acquisition. Four types of fibers (i.e., PVA, Spectra, Hooked,

and Torex) with volume fractions ranging from 0.75% to 2% were evaluated in this type

of test. Also, as for the dogbone tensile tests, a mortar compressive strength of 8.1 ksi, 56

MPa, was used throughout.

40

Figure 3.22: Crack opening displacement test set-up

Figure 3.23: Test setup configuration

3.6 Ring Tensile Test

In order to further observe the tensile response of HPFRCCs, an alternative

testing method, was considered. It consists of a ring which, when pressurized along its

inside surface, is subjected to an almost uniform tension. The reason this new method

was explored is to eliminate the drawbacks of other methods such as the dogbone tensile

specimen. Indeed, some of the problems encountered with tensile testing the dogbone

shaped specimens include the following:

1. Specimen construction problem: The wire meshes, which were used at the ends

for preventing local grip zone failure, are difficult to place in the specimen.

41

2. Bending problem: Eccentricity that occurs during loading due to crack formation

causes difficulties in analyzing the results.

3. Size problem: the gauge length where measurements can be made is small in

comparison to the total length of the specimen (in our case only 8/21 inches)

4. Testing problem: The direct tensile test is sensitive to many parameters, in

particular alignment and gripping conditions.

A ring-tensile test of concrete was then considered as an alternative testing

technique that would eliminate the above described problems.

The ring-setup was designed using thick steel plates as shown in Fig. 3.24. A

steel cone shaped wedge was manufactured that fits inside an inverse conical opening

formed by steel plates cut and assembled in the shape of pie slices. Driving the cone

inside the opening pushed the plates out, exercising pressure on the inside of the ring

specimen, thus creating the tensile stresses in the ring. The specimen tested was a

circular HPFRCC ring having an inside diameter equal to 304.8 mm, 12 inches and an

outside diameter equal to 406.4 mm, 16 inches. The specimen width was 50.8 mm, 2

inches, while the specimen’s depth was 76.2 mm, 3 inches.

Figure 3.24: Ring tensile test set-up

Figure 3.25: HPFRCC ring specimen

An Instron universal testing machine was used to apply the load vertically to the

conical prism drove in the wedge, which transformed the force horizontally to break the

specimen.

42

Figure 3.26: Ring HPFRCC specimen

Figure 3.27: Ring HPFRCC setup

A preliminary study was conducted to determine the most efficient number of

components needed to break the specimen. Three setups were considered; 2 pieces of

steel plates, 4 pieces of steel plates and 8 pieces of steel plates. Eight pieces were

subsequently selected to further study possible means for applying stress to the HPFRCC

specimen, using a finite element analysis. A cone wedge’s slope, θ = tan-1 (1.5/9) =

9.4623o was selected.

Figure 3.28: Interaction between steel plate and cone wedge setup

3.6.1 Result of Finite Element Analysis

Figure 3.29: Stress-contour of finite element analysis for two, four, and eight piece

steel-plate configurations

43

The result of the finite element analysis are illustrated in Fig. 3.29 where it is

clear that using the configuration with 8 steel plate pieces as a configuration leads to

minimal stress concentration when compared with the other two configurations. Thus the

8 steel plates configuration was selected.

The advantages of this test method are as follows:

1. The specimen does not require any internal specimen reinforcement.

2. Multiple cracking behavior is observable over a large specimen area, (i.e., the

entire specimen) compared to the gauge length test of the Dogbone specimen,

which can be seen only in the specimen’s middle portion.

3. The test remains very stable during loading, in comparison to a direct tensile

test.

Figure 3.30: Specimen setup Figure 3.31: Optotrak sensors

44

Figure 3.32: Testing machine and tensile

ring setup

Figure 3.33: Testing machine and tensile

ring setup

An Instron universal testing machine was employed in specimen testing. To begin

the process, monotonic loading was used on top of the cone wedge, which was inserted

into the steel plates exerting force upon the specimen. The force imposed upon the cone

wedge caused it to push through the hole inducing force on the steel plates, resulting in

expansion and specimen loading.

Three methods were utilized to measure the specimen’s displacement.

Figure 3.34: LVDT setup at specimen’s surface

Figure 3.35: LVDT setup at cone

wedge

The vertical displacement of the conical wedge was measured by an LVDT.

Furthermore, a spring type LVDT was connected by steel wire running along the outer

perimeter of the ring, to measure the total expansion.

45

A third method to measure displacement was used utilizing an Optotrak

instrument. The Optotrak camera system was placed on the table horizontally. Four

sensors were situated at the surface of the ring specimens to measure the surface level

displacement. The data was then used to estimate the total expansion of the ring under

load.

3.7 Data Acquisition System

Figure 3.36: Data collecting system for the direct tension tests

The experimental results from the direct tensile tests (dogbone test) and the crack

opening displacement tests (COD) are loads and related displacements. Testing data was

obtained from the experiment by using sensors (LVDTs etc.) connected to a National

Instrument DAC 6036E 16 Bits 16 channels 200/ms data acquisition card. The data was

subsequently stored in a laptop computer. The load versus displacement relationship was

then plotted using this data file. The Labview 7.1 data acquisition program was utilized to

preprocess the data. To achieve the highest accuracy level of the collecting data, the

sampling rate of data was set at 1000 points per second and the averaging process was

carried out by reducing the rate of sampling to 5 data per second. The AC/ LVDTs

sensors were attached to a Schaevitz signal conditioner to prepare the sensor’s signal for

appropriate voltage and response before processing the signal via the data acquisition

card. The load data and machine displacement position were also obtained from the

46

sensors in the MTS-810 universal testing machine and were linked to the data acquisition

card. By combining the load data from the load cell, and the displacement data from the

LVDTs, the load-displacement relation was presented in real-time and recorded during

the test.

Figure 3.37: Data acquisition system

Figure 3.38: View of real-time records of

stress-elongation response curves

A non-contact motion measuring instrument, called Optotrak was used in this

experimental program. Three infrared cameras from the Optotrak system were connected

to the computer running NDI software. Infrared sensors were attached to the specimen at

an observable predetermined gauge location. The displacements between two sensors

were recorded and used to compute strains. The movement of the infrared sensors

attached to the specimen was measured in the X, Y, and Z directions. The data of the

infrared sensors was linked to the data acquisition computer and was recorded at the same

time as the data from the machine load cell, machine displacement, and the two LVDTs

attached to the specimen. Thus, data derived from the four types of sensors were recorded

simultaneously.

47

3.8 Image Acquisition

Figure 3.39: Diagram for image acquisition system

A visualization system from National Instruments was used in this research. It

comprised an NI-1455 Compact Vision System with Labview RT 7.1 image processing

software. The image acquisition system was connected to a Sony XCD-710 digital

camera. The camera was equipped with a Nikkor 60mm f2.8 high definition macro lens.

The pictures taken from the digital camera were immediately linked to the experimental

data obtained from the load cell, machine displacement, and LVDT, and printed in the

images to show the status of testing at the time the pictures were taken. Consecutive

serial pictures of the test were kept and stored in the main memory of the NI-1455

Compact Vision System, and later on, exported to the main storage area in the data

acquisition computer. Synchronization of the data was performed in real-time and in an

effective manner. The data from the camera was reviewed and analyzed in depth. The

results from the analysis are described in Chapter 6.

48

Figure 3.40: Image acquisition processing

hardware

(NI-CVS1453)

Figure 3.41: Image acquisition system

3.9 Data Processing

Figure 3.42: Typical data and average curve

For comparison purpose, an average curve for each test series or parameter was

determined. The average plot was calculated according to an averaging procedure

initially described by Naaman and Najm (1991). The peak load of the average curve is

the average maximum load of the several individual tests, and their displacement is the

average corresponding displacement. The ascending and descending portions of each

individual curve are divided into a chosen number of points (100 points for this research)

49

and their corresponding loads and displacements recorded. The coordinates of the n th

point (from 1 to 100) of each curve are then averaged at their same rank to obtain the

average curve.

3.10 Concluding Remarks

This chapter covered the material properties and test procedures used in this study

and included a detailed description of constituent material properties, mixture

composition, fiber properties, specimen preparation, environmental conditions, testing

configuration, data acquisition, and the main parameters of the testing program. The

objective was to evaluate the behavior of HPFRCCs over the entire post-cracking range.

50

Table 3.4: Summary of testing program

51

CHAPTER 4

DIRECT TENSILE TESTS OF FIBER REINFORCED CEMENT COMPOSITES

4.1 Introduction

Several tensile test methods are available and have been used with fiber

reinforced cement composites. Each has its own advantages and disadvantages. Most

commonly, however, tensile testing is performed indirectly via flexural tests and split-

tensile tests. So far, direct tensile testing has been seldom used, yet it is one of the best

methods to characterize the tensile behavior of fiber reinforced cement composites. .

As previously described in Chapters 2-3, the direct tensile response of HPFRCC

can be viewed as the cumulative response of several cracks superimposed together. The

behavior of each cracked section can be clarified and modeled from the stress versus

crack-opening displacement test (Chapter 5). In this chapter, the direct tensile test results

from several series of specimens are presented. Fibers used included PVA fibers, Spectra

fibers, regular strength Hooked steel fiber, high strength Hooked steel fiber, regular

strength Torex steel fiber, and high strength Torex steel fiber. Load data from the load

cell of the testing machine were divided by the cross-sectional area of the specimen to

obtain stresses. Additionally, the average displacement of two LVDTs, placed on either

side of the specimen, was divided by the gauge length to obtain the strain at each stress

level, up to the peak load. In case of a single crack, the crack opening was calculated as

shown in Fig. 4.1b. An average curve was selected from each series of tests and used for

comparison and modeling.

52

.

Strain hardening composite Strain softening composite

(a)

(b)

Figure 4.1: Typical tensile behavior of (a) strain hardening composite and (b) strain

softening composite

P

L

ΔL

Crack width: LAPLLw Δ≈−Δ=

P

L

ΔL

Strain: LLΔ

=ε

53

Figure 4.2: Typical behavior of HPFRCC

The tensile response of all fiber reinforced cement composites can be classified

into two categories: a strain hardening composite or a strain softening composite. If the

specimen exhibits strain hardening composite, the pseudo strain up to the peak laod can

be calculated from the elongation (ΔL) divided by the total gauge length (L). However, if

the specimen exhibits strain-softening behavior (e.g., FRC, or HPFRCC after the

specimen reaches the localization phase, or the crack opening phase), crack width can be

calculated from the elongation of the specimen. The behavior in the strain softening

stage results from the pull out behavior and/or rupture of fibers interacting with the

cementitious material. Typically, the strain softening behavior occurs after the specimen

reaches its maximum tensile resistance (Fig. 4.1).

Fiber pull-out

Fiber

failure

Possible

localization

range

Multiple

cracking stage Maximum post-cracking stress

A

B

C

Stress

Strain

(Displacement)

54

Figure 4.3: Strain at maximum stress and strain at the end of multiple cracking stages

Typically, the area under the stress-strain curve represents the energy absorbed by

the specimen during direct tensile testing, and the area under the curve after localization

represents the surface energy of the composite. In this chapter, the energy (area under the

curve) of the specimens up to first cracking stress, maximum post-cracking tensile stress,

and at the end of the multiple cracking stages, were analyzed and investigated. The point

representing the end of multiple cracking was introduced because in many tests some

additional cracks occurred after the peak load, and thus the strain or deformation at the

peak load did not fully represent the end of multiple cracking. Figure 4.3 illustrates a

typical such response.

4.2 Direct Tensile Behavior of Mortar without Fibers

Generally, the tensile strength of cement mortar (without fiber) is approximately

10% of its compressive strength. Table 4.1 illustrates the tensile strength of mortar as a

function of its compressive strength.

55

Table 4.1: Common mechanical properties of mortar

Common design value Property Observed range

U.S. units S.I. units

Direct

tensile

strength,

f’ct

'' 53 cc ftof −− ''

313 ccc forf γ−− '' 0069.025.0 ccc forf γ−−

Split

cylinder

tensile

strength

'' 76 cc ftof −− '' 6.06 ccc forf γ−−

'' 00138.050.0 ccc forf γ−−

Modulus

of

rupture fr

'' 125.7 cc ftof −−

'5.7 cf− '62.0 cf−

Poisson’

s ratio, υ 0.15 to 0.20 0.20 0.20

Table 4.2: Identification of mortar specimens and mortar composition

Identification

of specimen

Type of

specimen

Number

of

specimen

Type of

matrix

Type of

fibers

Volume

of fibers

Vf(%)

D-N-0 Dogbone 7 mortar NA NA

In order to provide a basis of comparison and for control, the direct tensile

strength of mortar was investigated. Seven mortar specimens, without fiber, were tested

using dogbone specimens with same dimensions as the fiber reinforced specimens and

same mortar composition (Table 4.2). They were identified as series D-N-O (Table 4.2),

Results of the experimental tests are summarized in Table 4.3 and 4.4, and include

the maximum stress (or first cracking stress), the corresponding strain, and the elastic

56

modulus of the material. The stress-strain curves recorded are plotted in Fig. 4.4; photos

of a typical specimen during testing are shown in Fig. 4.5, and photos of specimens after

testing are shown in Figs. 4.6 and 4.7.

Theoretically, the tensile response of mortar should exhibit linear elasticity up to

the first cracking point, then fail by the opening of the first crack. However, from the

experimental observations (Fig. 4.5), the stress-strain behavior of the mortar specimens

did not show perfect linear elastic behavior. Indeed beyond about 70% of the maximum

load, the response shows an inelastic behavior with a decreasing slope. This may be

attributed to the slow progress of the critical crack along the critical section prior to

localization, thus leading to a reduction in cross-sectional area and stiffness. In Fig. 4.5b,

the initiation of the critical crack can be observed on the right lower side of the specimen.

Until this crack reaches the other side, a reduction in stiffness observed until brittle

failure occurs.

The elastic modulus measured at (10% to 50% of fc) ranged between 1132 ksi to

3875 ksi, (7804 to 26717 MPa) with an average of 2014 ksi, (13886 MPa). Note as

expected, the mortar appears to have significantly less stiffness than regular concrete with

same compressive strength (3500-4500 ksi, 24131-31026 MPa). The lower stiffness

results from a lack of coarse aggregates in the mortar composition.

Note that the cracking strength observed ranged from 103.3 psi to 281.6 psi,

(0.712 to 1.941 MPa) with an average of 181.52 psi (1.25 MPa). However, the variability

was also high. The coefficient of variation is approximately 50.08% for the maximum

stress, and 53.48% for the modulus of elasticity. This large variability is due to

sensitivity of direct tensile testing of brittle materials like mortar (weakest link effect), the

gripping conditions, and the variation due to mixing and curing. The approximate direct

tensile strength from this test is around '2 cf psi, which is lower than typically found in

a standard ASTM mortar test utilizing a briquette )53( '' psiftof cc − . The average

observed strain at failure was at 0.000196, and its coefficient of variation was higher than

found in maximum stress and the modulus of elasticity (68.41%).

All specimens failed in a brittle manner. Only a single crack was observed in

each specimen. With most specimens, the failure crack occurred in the middle portion of

57

the specimens, that is, within the gauge length.

Table 4.3: Direct tensile test of mortar (US-units)

Identification of

specimen Cast date Test date

Maximum

stress (psi)

Modulus of

elasticity (ksi) Strain at failure

D-N-O-1 6/21/2006 9/12/2006 132.75 1610 0.00011

D-N-O-2 6/21/2006 9/13/2006 224.12 1987 0.00025

D-N-O-3 6/21/2006 9/12/2006 103.33 2045 0.0001

D-N-O-4 6/18/2006 9/13/2006 130.94 1759 0.00025

D-N-O-5 6/18/2006 9/13/2006 127.38 1694 0.00015

D-N-O-6 6/18/2006 9/12/2006 270.50 1132 0.00045

D-N-O-7 3/12/2006 2/16/2007 281.60 3875 0.00006

Average 181.52 2014.57 0.000196

Standard deviation 94.54 1077.35 0.000134

Coefficient of variation (COV) (%) 52.08 53.48 68.41

Table 4.4: Direct tensile test of mortar (SI-units)

Identification of

specimen Cast date Test date

Maximum

stress (MPa)

Modulus of

elasticity

(MPa)

Strain at failure

D-N-O-1 6/21/2006 9/12/2006 0.915 11101 0.00011

D-N-O-2 6/21/2006 9/13/2006 1.545 13700 0.00025

D-N-O-3 6/21/2006 9/12/2006 0.712 14100 0.0001

D-N-O-4 6/18/2006 9/13/2006 0.903 12128 0.00025

D-N-O-5 6/18/2006 9/13/2006 0.878 11680 0.00015

D-N-O-6 6/18/2006 9/12/2006 1.865 7805 0.00045

D-N-O-7 3/12/2006 2/16/2007 1.942 26717 0.00006

Average 1.252 13890 0.000196

Standard Deviation 0.652 7428 0.000134

Coefficient of Variation (COV) (%) 52.08 53.48 68.41

58

Figure 4.4: Testing results of specimen without fiber (control specimen)

59

(a) Initial status

Stress = 29.75 psi

(0.205 MPa) No observable crack.

(b) Maximum stress status

Stress = 280.243 psi

(1.932 MPa)

Small visible crack at the

lower part of specimen.

(c) Failure

Stress = 0 psi, (0 MPa)

Completed crack.

Brittle failure.

Figure 4.5: Testing of typical specimen without fiber (D-N-O-7), (a) initial status, (b)

maximum stress status, and (c) failure

Figure 4.6: Dogbone specimen without fibers (specimen 1, 2, 3)

60

Figure 4.7: Dogbone specimen without fibers (specimen 4, 5, 6)

4.3 Direct Tensile Behavior of FRCC Reinforced with PVA Fibers

Twenty five direct tensile specimens of FRCC reinforced with PVA fibers were

tested and can be classified into two categories, namely FRCC reinforced non oiled PVA

fiber (PVA-L), and FRCC reinforced oiled PVA fiber (PVA-H). A summary of test

details as well as specimen identification are given in Table 4.5.

The following are the mechanical properties of the PVA fiber as supplied by the

manufacturer: Diameter = 0.19 mm -> 7.48x10-3 inch

Length = 12 mm -> 0.5 inch

Frictional bond strength τ = 3.5 MPa ->510 psi (Guerrero, 1999)

Aspect ratio: l/d = 66.84

61

Table 4.5: Summary of direct tensile test series with PVA fiber

The following identification is used (Table 4.5). The first letter denotes the type

of test (D for direct tensile test, Dogbone test). The second letter denotes the type of fiber

(P = PVA fiber). The third letter denotes the type of fiber (H = oiled PVA type, L = non-

oiled PVA type). The fourth letter denotes the volume fraction of fiber, (that is, for the

oiled fibers, 2.0%, 1.5%, 1.0%, 0.75%).

4.3.1 Results of Test Series with PVA-H Fiber

The tensile stress-strain curves (where the strain is valid up to the peak stress

only) for the test series with oiled PVA fibers are given in Figs. 4.8 (a, b, c, d), and are

discussed below. Note that each graph shows two curves and their average.

62

(a) (b)

(c) (d)

Figure 4.8: Stress-strain curves of specimens reinforced with PVA-H (D-P-H),

(a) Vf = 0.75%, (b) Vf = 1.0%, (c) Vf = 1.5%, and (d) Vf = 2.0%

All specimens were initially pre-loaded to around 100 lbs (50 psi, 0.34 MPa) to

eliminate possible errors from grip adjustment and settlement of support. The loading

rate was 0.02 inch / minute (0.508 mm / minute); then the load increased with increased

displacement.

It can be observed that the initial portion of the curves is almost linearly elastic up

to the first cracking, which occurred at approximately 400-500 psi, (2.75-3.45 MPa).

This value of the first cracking stress is nearly 2.5 times higher than the cracking stress of

specimens without fiber (181.52 psi, 1.25 MPa); this indicates the benefits of PVA fiber

in improving the cracking resistance. The strain (0.00016) at the elastic limit is almost the

same as for the specimens without fiber.

63

Some inelastic behavior of specimens with PVA fibers could be observed as the

first crack initiated from one side of the specimen and propagated slowly to the other

side. After first cracking, the stress dropped, and the specimen changed stiffness. Some

specimens experienced strain hardening behavior, due to some fiber pullout along the

critical crack. However, some specimens exhibited some ductility (say up to a strain of

0.008) before tensile softening (PVA 2.0%).

Figure 4.9: Crack in direct tensile

specimen reinforced with PVA fiber

Only a single crack was observed in each

specimen. All specimens showed clear, full,

non-jagged cracks. A slight damage in the

specimen’s matrix at the crack due to fiber pull

out was observed from analyzing the cracked

surfaces. Most of the fibers broke almost

uniformly at a crack opening of about 0.1 inch

(2.54 mm). Figure 4.9 illustrates a typical

failure crack.

Figure 4.10: Comparison of average curves for specimens reinforced with PVA-H fiber

with different volume fractions

64

Figure 4.10 compares the average curves for all the test series with PVA oiled

fibers. It can be observed that an increase in fiber volume fraction from 0.75% to 1.5%

leads to an increase in maximum stress, but no increase in strain at maximum stress. For

2% fiber content, the maximum stress is lower than for 1.5% indicating possible effect of

difficulties due to mixing. The strain at maximum stress was highest for the series with

2% fibers and close to 1/1000.

Figure 4.11 (a to d) illustrate the variation of various test results with the fiber

volume fraction. No strong trend could be detected except that the energy at first

cracking can vary widely, indeed due to the nature of first cracking.

(a)

(b)

(c)

(d)

Figure 4.11: Comparison of different properties of test series with oiled PVA fibers at

different fiber volume fractions, (a) first cracking stress, (b) energy at first cracking

stress, (c) maximum stress, and (d) energy at post cracking stress

65

Table 4.6 Test results of direct tensile specimens reinforced with PVA-H fiber (US-units)

Table 4.7 Test results of direct tensile specimens reinforced with PVA-H fiber (SI-units)

66

Figure 4.12: Specimens reinforced with PVA-H after testing (D-P-H)

4.3.2 Results of Test Series with PVA-L Fiber

Seventeen specimens reinforced with PVA-L fiber (non-oiled type PVA) were

tested (Table 4.5, D-P-L), having a fiber volume fraction of 1.0%, 1.5%, and 2.0%. For

each volume fraction, 5 to 6 specimens were tested. Average stress-strain curves, where

strain is valid up to peak stress only, are given in Fig. 4.13.

Figure 4.13: Comparison of tensile response of specimens with PVA-L fiber at different

fiber volume fraction

67

Three key observations can be made: 1) the maximum stress (from about 150 psi

to 200 psi) in all these series is significantly smaller than for the series with oiled PVA

fibers; 2) an increase in fiber volume fraction leads to a decrease in maximum stress; and

3) the strain at maximum stress is approximately 0.2% for the specimens with oiled

fibers. These results are difficult to explain and can be attributed in part to the increased

difficulty to mix higher volume fractions of fibers without changing the mix proportions

and composition, thus leading to deterioration in properties. Even the first observation

where the cracking strength with non-oiled fibers is smaller than that with oiled fiber

does not follow logic.

(a) (b)

Figure 4.14: Typical failure sections of specimens reinforced with (a) non-oiled PVA-L

fibers and (b) oiled PVA-H fibers

To further understand the above results, typical cross-sectional areas of cracked

sections were analyzed. An example is shown in Fig. 4.14. It was observed that the

specimens with non-oiled fibers had lumps or groups of fibers stuck together while the

specimens with oiled fibers had a more uniform distribution of fibers. This difference

may very well explain the above unexpected observations.

If one ignores the above analysis, then it can be said that the use of oiled fibers

leads to a better tensile resistance and a better energy absorption capacity than the use of

non-oiled fibers.

68

Table 4.8: Summary of test results of specimens reinforced PVA-L fiber (US-units)

69

Table 4.9: Summary of test results of specimens reinforced with PVA-L fiber (SI-units)

4.3.3 Concluding Remarks

1. No clear multiple cracking behaviors were observed in specimen reinforced with

PVA fiber, whether the fibers were oiled or not. By and large, one crack was

observed in all tests. This may seem in conflict with findings from other

investigators, but could be attributed to the fact that they used different methods

of tensile testing and smaller size specimens.

2. The presence of PVA fiber effectively improves the cracking stress, and the

cracking strain of the matrix. However, comparing improvement with other types

of fiber, such as Spectra, Hooked and Torex, the effectiveness of PVA fiber was

the lowest.

70

3. Increasing the volume fraction of oiled PVA fibers up to 1.5% by volume, led to a

marked improvement in the post-cracking strength, ductility, and energy

absorption capacity of the composite.

4. Given the specimen preparation and testing procedure used, the optimum volume

fraction of fiber was close to 1.5%. The highest direct tensile stress observed from

specimens reinforced with PVA fibers was 482 psi (3.323 MPa) at 1.5% volume

fraction.

5. Because of the difficulty encountered in mixing non-oiled PVA fibers, no

particular logical conclusion could be drawn from the related tests.

4.4 Direct Tensile Behavior of HPFRCC Reinforced with Spectra Fibers

Forty two direct tensile specimens of HPFRCC reinforced with Spectra fibers

were tested. A summary of test details as well as specimen identification are given in

Table 4.10.

The following are the mechanical properties of the PVA fiber as supplied by the

manufacturer:

An identification was given to each series where the first letter denotes the type of

test (D for direct tensile test, dogbone test), the second letter denotes the type of fiber (S

= Spectra fiber), and the third letter denotes the volume fraction of fiber, (2.0%, 1.5%,

1.0%, or 0.75%). Followings are the mechanical properties of the Spectra fiber as

supplied by the manufacturer and from prior tests:

Diameter = 0.038 mm -> 1.49x10-3 inch


Frictional bond strength τ = 0.62 MPa ->89.923 psi (Li and Lieung, 1991)

Aspect ratio: l/d = 1000

71

Table 4.10: Summary of HPFRCC reinforced Spectra test

Identification

of Specimen

Type of

Specimen

Number of

Specimens

Type of

Matrix

Type of

Fibers

Volume

of

Fibers

Vf (%)

D-S-2 Dogbone 17 Mortar Spectra 2.00%

D-S-1.5 Dogbone 13 Mortar Spectra 1.50%

D-S-1 Dogbone 11 Mortar Spectra 1.00%

D-S-0.75 Dogbone 1 Mortar Spectra 0.75%

(a)

(b)

(c)

(d)

Figure 4.15: Stress-strain curves of specimens reinforced with Spectra (D-S) fiber at

volume fractions of: (a) 0.75%, (b) 1%, (c) 1.5%, and (d) 2%

Average

72

The actual and average stress strain curves, where strain is valid up to peak stress

only, are shown in Fig. 4.15 for the four volume fractions of fiber used. It can be

observed that all specimens behaved linearly up to first cracking with an elastic modulus

of the same order about 2000 ksi or13890 MPa. The first cracking stress was around 300

psi (2.068 MPa). The large number of tests carried out at 1%, 1.5% and 2% illustrates

the large variability that can be observed in this type of tensile testing. Variability will be

discussed later in Chapter 7.

Figure 4.16: Comparison of average curves for specimen reinforced with Spectra fiber at

different fiber volume fractions

Figure 4.16 compares the average stress strain curves from the tests. The first

cracking stress is nearly comparable for every volume fraction of fiber, and

approximately equal to 300 psi, (2.068 MPa). In all cases, following first cracking,

multiple cracking and strain hardening was observed. Overall, increasing the volume

fraction of fibers leads to an increase post-cracking tensile strength, ductility and energy

absorption capacity.

Note that the ductility in tension of specimens with Spectra fibers was

exceptionally good when a comparison is made with PVA fibers (compare the x axis

scale). Beyond the peak stress, the softening portion is gradual and exhibits mostly fibers

pulling out from the matrix and breaking of the matrix.

73

Figures 4.17, 4.18, and 4.19 show the variation of test results related to first

cracking in terms of the volume fraction of fibers. While the average first cracking stress

ranged from 262 psi (1.806 MPa) to 328 psi (2.261 MPa), it can be said from Fig. 4.17

that the volume fraction has little influence on the cracking strength. Similarly, no clear

trend could be observed for the strain and the energy absorption capacity at cracking

(Figs. 4.18, 4.19). Moreover, the first cracking strength with Spectra fibers was

consistently higher than that of specimens without fiber (i.e., 44% to 81% higher).

The strain at first cracking stress ranged from 0.00082-0.00176, that is,

significantly higher than that of specimens without fibers (Strain average = 0.000196.)

Figure 4.17: First cracking stress versus

volume fraction

Figure 4.18: Strain at first cracking stress

versus volume fraction

Figure 4.19: Energy at first cracking stress versus volume fraction

74

Figures 4.20, 4.21, and 4.22 show the variation of test results related to maximum

post-cracking stress point in terms of the volume fraction of fibers. The strain at

maximum stress was significantly higher than the strain at first cracking stress. In the

multiple cracking stages, the number of cracks increased with increasing elongation. The

average maximum stress ranged from 299 psi (2.062 MPa) to 466 psi (3.212 MPa) (Fig.

4.20). The ratio of maximum stress to first cracking stress is presented in Table 4.11.

Table 4.11: Volume fraction of fiber and the stress ratio

Volume

fraction 0.75% 1.00% 1.50% 2.00%

cc

pc

σσ

1.14 1.03 1.36 1.38

Moreover, the strain at maximum stress, and the energy at maximum stress seem

to increase with the volume fraction of fiber up to 1.5%, but decrease thereafter at 2%

fiber content (Figs. 4.21 and 4.22). This is surely due to the fact that the higher the fiber

content, the more difficult it is to mix the fibers, thus leading to air entrapment and poorer

properties.

Figure 4.20: Specimen at maximum stress Figure 4.21: Strain at maximum stress

75

Figure 4.22: Specimens’ energy at maximum stress

Since the peak point did not represent in all cases the end of multiple cracking,

another point defined as the end of multiple cracking points was also identified. It

generally occurred at about 80% of the maximum load on the softening branch of the

curve. Related results are described in Figs. 4.23 and 4.24. These figures show an

increase of strain and energy absorption capacity with volume fraction of fibers, but up to

a certain level where deterioration may follow.

Figure 4.23: Variation of strain at the

end of multiple cracking

Figure 4.24: Variation of energy at the end of

multiple cracking

76


1. The use of Spectra fibers lead to a marked improvement in specimen ductility,

energy absorption capacity, strain hardening response and the extent of multiple

cracking. Increasing the volume fraction of fibers up to 1.5% generally led to

improvement in properties. However, at 2% fiber content, properties started

deteriorating due to difficulties in mixing and related air entrapment.

2. Increasing the volume fraction of Spectra fibers up to 1.5% by volume led to an

increase in number of cracks and a decrease in crack width and spacing.

3. A consistent correlation could be established between the volume fraction of

fiber and the first cracking stress (approximately 300 psi, 2 MPa), the strain at

first cracking, and the corresponding energy absorption capacity.

4. Increasing volume fraction of fiber showed overall improvement in the maximum

post-cracking stress, the strain at maximum stress, and related energy at the end of

multiple cracking or crack saturation.

5. Everything else being equal and given the parameters of this study and the

Spectra fiber used (length and diameter) the optimum volume fraction of Spectra

fiber seems to be around 1.5%.

77

Table 4.12: Summary of test results for specimens reinforced with Spectra fiber

(US-units)

78

Table 4.13: Summary of test results for specimens reinforced with Spectra fiber (SI-units)

79

4.5 Direct Tensile Behavior of HPFRCC Reinforced with Hooked Steel

Fiber

In all sixty three specimens were tested; the test series were given an

identification name, and classified into two categories, one dealing with high strength

steel Hooked fiber (D-H-H) and the other with regular strength steel hooked fiber (D-H-

L). Table 4.14 describes the various test series and their identification. The first letter

denotes the type of test (D for direct tensile test, Dogbone test). The second letter denotes

the type of fiber (H = Hooked fiber), and the third letter denotes the type of Hooked fiber

(H = High strength Hooked type, L = Regular strength Hooked type). Furthermore, the

fourth letter denotes the volume fraction of fiber, (2.0%, 1.5%, 1.0%, or 0.75%).

Following are the main properties of the fibers used:

Hooked - High Strength

Diameter = 0.4 mm -> 0.0157 inches

Tensile strength -> 2100 MPa


Equivalent bond strength τ = 5.1 MPa ->740 psi (Guerrero, 1998) Aspect

ratio l/d = 75

Color = gold

Hooked - Regular Strength


Tensile strength -> 1050 MPa


Equivalent bond strength τ = 5.1 MPa ->740 psi (Guerrero, 1998)

Aspect l/d = 100

Color = silver

80

4.5.1 Result of Test Series with High Strength Hooked Steel Fiber

Forty-six specimens reinforced with high strength Hooked fiber were tested. Four

volume fractions were used (0.75%, 1.0%, 1.50%, and 2.0%).

Table 4.14 Identification of specimen reinforced Hooked fiber

Figures 4.25a to 4.25d give the stress-strain curves (where strain is valid up to

peak stress only) for each series of specimens, and the series average curve, at the four

volume fractions of fiber used. A great variability is generally observed and is further

discussed in Chapter 7.

81

(a) (b)

(c)

(d)

Figure 4.25: Stress strain curves of HPFRCC reinforced with high strength Hooked steel

fiber at different volume fractions of fiber and average curves, (a) Vf = 0.75%,

(b) Vf = 1.0%, (c) Vf = 1.5%, and (d) Vf = 2.0%

82

Figure 4.26: Average stress strain curves of HPFRCC reinforced with high strength

Hooked steel fiber

Figure 4.26 provides a comparison of the average curves at the four volume

fractions of fiber tested. The main test results are summarized in Tables 4.15 and 4.16. It

can be generally observed that an increase in fiber volume fraction leads to an

improvement in the stress-strain response.

Variation of stress, strain and related energy at the first cracking point are

illustrated in Figs. 4.27 to 4.29. They do not show a consistent trend; indeed if the data

for Vf= 0.75% is ignored, one can conclude that the stress at first cracking does not

change much for Vf = 1%, 1.5% and 2%. Related increases in strain, thus energy, may be

attributed in part to the nonlinearity of the curves prior to the first through-specimen

crack and in part to variation in the measurements as observed from the large variability

encountered (Fig. 4.25). Initially, the specimen exhibits linear elastic behavior up to the

first cracking stress. Several events occurred at this point such as matrix breaking, crack

propagation, and fiber shape alteration. The average first cracking stress ranged from 158

psi (1.089 MPa) to 290 psi (2 MPa).

83

Figure 4.27: Stress at the first cracking

versus fiber volume fraction

Figure 4.28: Strain at the first cracking


Figure 4.29: Energy at first cracking versus fiber volume fraction

After the first cracking, most specimens exhibited tensile hardening behavior with

multiple cracking up to maximum stress. . At localization just after maximum stress,

there is evidence of pull-out and damage with in the critical section.

Variation of stress, strain and related energy at the maximum post-cracking point

are illustrated in Figs. 4.30 to 4.32. They show that the maximum stress and the energy at

maximum stress increase significantly with an increase in fiber volume fraction, while

the strain at maximum stress remains almost the same and of the order of 0.4%. The

maximum tensile resistance in HPFRCC reinforced Hooked steel fiber is considerably

better than that obtained from HPFRCC reinforced polymetric fibers (PVA and Spectra).

84

However, the strain at maximum stress, while much larger than for specimens with PVA

fiber, is smaller than for specimens with Spectra fiber.

Figure 4.30: Maximum stress versus fiber

volume fraction

Figure 4.31: Strain at maximum stress


Figure 4.32: Energy at maximum stress point versus fiber volume fraction

Because some multiple cracking continued beyond the maximum stress point, an

additional point defined as the end of multiple cracking on the softening branch following

the maximum stress point, was analyzed. Figures 4.33 and 4.34 illustrate the

characteristics of this point versus the fiber volume fraction. It can be observed that the

strain at the end of multiple cracking point remains about constant at 0.6% when the

volume fraction of fibers increases from 0.75% to 2%, while the related energy increases

since the corresponding stress also increases.

85

Figure 4.33: Strain at the end of multiple

cracking versus fiber volume fraction

Figure 4.34: Energy at the end of multiple

cracking versus fiber volume fraction

86

Table 4.15 Summary of test results for specimens reinforced with high strength Hooked

steel fiber (US-units)

87

Table 4.16: Summary of test results for specimens reinforced with high strength Hooked

steel fiber (SI-units)

88

4.5.2 Result of Test Series with Regular Strength Hooked Steel Fiber

Seventeen specimens reinforced with regular strength Hooked steel fiber were

tested at three volume fractions of fiber (1.0%, 1.5% and 2.0%). Their average stress-

strain curves (where strain is valid up to peak stress only) are compared in Fig. 4.35. A

summary of the test results is given in Tables 4.17 and 4.18.

Figure 4.35: Average stress strain curves of HPFRCC reinforced with regular strength

Hooked steel fiber at different fiber volume fractions

The results plotted in Fig. 4.35 seem to confirm the same trends observed when

using high strength hooked steel fiber: that is an increase in maximum stress and related

energy with an increase in volume fraction of fibers, while the strain at maximum stress

remains almost the same. In all tests, localization occurred while the hooked fibers

pulled out from the critical crack with no evidence of fiber failure. In comparing Figs.

4.35 and 4.32, it can be observed that both the stress and strain at peak load for the

89

regular strength hooked fiber were almost half those of the high strength hooked fiber.

In particular, the strain at maximum stress was of the order of 0.2% for the test series

with regular strength hooked fiber.

Table 4.17: Summary of test results for specimens reinforced with regular strength

Hooked steel fiber (US-units)

90

Table 4.18: Summary of test results for specimens reinforced with regular strength

Hooked steel fiber (SI-units)


Two types of hooked fiber were used, one made with high strength steel wire (of

tensile strength 2100 MPa) and the other with conventional steel wire (of tensile strength

1050 MPa).

1. The use of high strength versus regular strength Hooked steel fiber clearly

improved the strain hardening and multiple cracking behavior, toughness, and

energy absorption capacity. Most specimens achieved strain hardening response

even at 0.75% fiber content by volume.

91

2. Increasing the fiber volume fraction led to improvement in all properties except

for the stress and strain at first cracking, and the strain at maximum post-cracking

which did not clearly follow the trend.

3. Overall the higher strength fiber let to a better performance; for instance the

maximum tensile stress with the high strength fiber was 1.5 to 2.5 times that with

low strength fiber, at Vf of 1% to 2%.

4.6 Direct Tensile Behavior of HPFRCC Reinforced with Torex Twisted

Steel Fiber

The test series in this part of the program comprised 84 specimens and are

classified into two categories, namely specimens reinforced with high strength Torex

steel fiber (D-T-H), and specimens reinforced low strength Torex steel fiber (D-T-L).

Table 4.19 gives a summary of the test series and their identification. The first letter in

the identification denotes the type of test (D for direct tensile test, Dogbone test). The

second letter denotes the type of fiber (T = Torex fiber), and the third letter denotes the

type of Torex fiber (H = High strength steel Torex, L = regular strength steel Torex). The

fourth letter denotes the volume fraction of fiber, (2.0%, 1.5%, 1.0%, or 0.75%).

Following is a summary of the fiber properties after twisting

Torex - High Strength



Equivalent bond strength τ = 6.84 MPa ->992.28 psi (Sujivorakul,2002)

l/d = 100, Tensile strength = 2760 MPa

Color = gold

Torex- Regular Strength



Equivalent bond strength τ = 6.84 MPa ->992.28 psi (Sujivorakul,2002)

l/d = 100, Tensile strength = 1380 MPa

Color = silver

92

Table 4.19: Identification of specimen reinforced Torex fiber

4.6.1 Result of Test Series with High Strength Torex Steel Fiber

In this category (D-T-H), sixty-seven specimens reinforced with high strength

Torex steel fiber were tested. Each series used 5 to 28 specimens. Test results are plotted

in Figs. 4.35 to 4.40 and summarized in Tables 4.20 and 4.21.

Figures 4.36a to 4.36d show the stress strain curves observed from the tests

(where strain is valid up to peak stress only) and their average at each volume fraction of

fiber tested. Since the axis scales are different, a comparison of average curves is given

in Fig. 4.37. Similarly to the direct tensile tests with other fibers, a great variability was

observed (Fig. 4.36) and is further discussed in Chapter 7.

93

(a) (b)

(c) (d)

Figure 4.36: Stress strain curves of HPFRCC reinforced with high strength Torex steel

fiber at different volume fractions of fiber and average curves. (a) Vf = 0.75%,

(b) Vf = 1.0%, (c) Vf = 1.5%, and (d) Vf = 2.0%

94

Figure 4.37: Average stress strain curves of HPFRCC reinforced with high strength

Torex steel fiber at different fiber volume fractions

From the average curves in Fig. 4.37, it can be stated that the greater the fiber

volume fraction, the better the performance of the specimen in terms of stress, ductility,

and energy absorption occurring. Differences between results at 0.75% and 1% fiber

content were very small and attributable to normal variability.

Figures 4.38 to 4.40 describe the variation stress, strain and corresponding for the

point at first cracking. Generally, they all increase with an increase in volume fraction of

fiber, showing a slightly different trend than the case with Hooked fiber. The average

first cracking stress ranged from 251 psi to 357 psi, (1.731 to 2.461 MPa) (and the strain

at first cracking point ranged from 0.00012 to 0.000268.

95

Figure 4.38: First cracking stress versus

fiber volume fraction

Figure 4.39: Strain at first cracking stress


Figure 4.40: Energy at first cracking stress versus fiber volume fraction

All HPFRCC reinforced with Torex fiber exhibited good multiple cracking with

very well distributed cracks along the specimen length. Figures 4.41 to 4.43 describe the

variation of properties at the maximum post-cracking stress versus the volume fraction of

fiber. It can be generally concluded from the figures that the maximum stress, the strain

at maximum stress and the corresponding energy all increase with an increase if fiber

volume fraction.

The maximum tensile resistance of HPFRCC reinforced with high strength Torex

steel fiber ranged from 533 psi to 939 psi (3.675 to 6.474 MPa) while the strain at

maximum stress range from 0.2% to 0.4%. This strain was smaller than expected and as

96

observed in other investigations, but is believed due to the twisting ratio of the fiber

which was not optimized for the matrix used.

Figure 4.41: Maximum stress versus fiber

volume fraction

Figure 4.42: Strain at maximum stress


Figure 4.43: Energy at maximum stress versus fiber volume fraction

Since multiple cracking continued beyond the peak stress, a point indicating the

end of multiple cracking was selected. Figs. 4.44 and 4.45 describe the variation of strain

and energy absorption capacity at this point versus the fiber volume fraction. Now the

strain increases up to about 0.5% suggesting significant ductility.

97

Figure 4.44: Strain at the end of multiple

cracking stage versus fiber volume

fraction


cracking stage versus fiber volume fraction

98

Table 4.20: Summary of test results of specimen reinforced high strength Torex fiber

(US-units)

99

Table 4.21: Summary of test results of specimen reinforced high strength Torex fiber

(SI-units)

100

4.6.2 Result of Test Series with Regular Strength Torex Steel Fiber

Seventeen specimens reinforced with regular strength Torex steel fiber were

tested at three different fiber contents (1.0%, 1.5%, and 2.0%). It should be noted that

these fibers encountered many flaws during fabrication which included a non-uniform

shape, unequal twists and larger distance between ribs. These flaws were uncovered

during examination under a light microscope. The test results are shown below for

interest, but should not be considered typical since the fibers were faulty and should have

been rejected. The average stress-strain curves are plotted in Fig. 4.46 and a summary of

test results is given in Table 4.22 and 4.23. They are self-explanatory and, unfortunately,

cannot be used to draw meaningful conclusions except that, overall, they show a

performance lower than the series with high strength Torex fiber. During testing the

fibers pulled out from the critical section after localization and no fiber failure was

observed.

Figure 4.46: Average stress strain curves of HPFRCC reinforced with regular strength

Torex steel fiber at different fiber volume fractions

101

Table 4.22: Summary of test results of specimens reinforced regular strength Torex fiber

(US-units)

102

Table 4.23: Summary of test results of specimens reinforced regular strength Torex fiber

(SI-units)


1. The use of high strength Torex steel fiber consistently improved the strain

hardening and multiple cracking behavior, the maximum post-cracking stress, the

strain and energy at maximum stress, and related energy absorption capacity. The

higher the volume content of fibers, the higher the improvements observed in the

tensile behavior.

2. The number of cracks observed generally increased with the volume fraction of

fiber, leading to a decrease in crack spacing and width. Specimens with high

103

volume fraction of fiber also showed large ductility and energy absorption

capacity.

3. Two types of Torex fiber were used in this study, one made with high strength

steel wire (of tensile strength 2760 MPa), and one with regular strength steel wire

(of tensile strength 1380 MPa). The higher strength fiber led to a 30% to 60 %

better performance, in terms of maximum stress.

4.7 Comparison Between HPFRCC with Different Fibers

(a) (b)

(c)

(d)

Figure 4.47 Comparison of average stress-strain response of FRCC test series with

different fibers at fiber volume fractions of: (a) 0.75%, (b) 1%, (c) 1.5%, and (d) 2%

104

Figures 4.47 (a, b, c, d) compare the response of test series with different types of fiber at

the four values of fiber content used. It is clear from the figures that, everything else

being equal, the use of PVA fibers leads to the lowest performance at every volume

fraction in comparison to specimens reinforced with Spectra, Hooked or Torex fibers.

Also, specimens with Torex fibers lead to the highest post cracking strength while

specimens with Spectra fibers lead to the highest strain at maximum stress. At 2% fiber

content, specimens with Spectra fiber showed high ductility with a post-cracking tensile

strain of up to 2%, almost three to four times the strains observed with Hooked and Torex

fibers. However, this came at a much lower tensile stress. At 2% fiber content,

specimens with either high strength Hooked or Torex fibers lead to post-cracking tensile

strengths at least twice those observed for Spectra and PVA fibers. If the overall

performance of the four fibers is ranked, then the order would be Torex, Hooked, Spectra

and PVA, from highest to lowest.

4.8 Concluding Remarks on the Direct Tensile Tests of FRC Composites

with Different Fibers

This chapter addressed the direct tensile testing results of HPFRCC. A detailed

description of material properties was given and experimental results reported. Based on

the results obtained the following conclusions can be drawn:

1. Generally, the first cracking stress of the composite is significantly improved due

to the presence of fiber in the matrix. However, no consistent correlation could be

established between an increase in volume fraction of fiber and an increase in first

cracking stress. Given the parameters of this study, for Hooked and Torex fiber,

the stress at first cracking increased with the volume fraction, while it remained

almost same for PVA and Spectra fibers

2. Comparing between types of fiber, for the same volume fraction of fibers,

specimens with Torex, high strength Hooked, and Spectra fibers showed better

overall behavior than specimens with PVA fibers. Moreover, their post-cracking

strength reached 1.5 to 3 times their strength at first cracking, while for specimens

105

with PVA fibers, the post-cracking strength was only 10% to 50% higher than the

cracking strength at up to 2% fiber content.

3. The tensile response of specimens without fibers is very brittle and shows large

variability in tensile strength. Fibers, whether leading to strain softening or strain

hardening response, reduce the variability while improving ductility and

toughness.

4. Immediately following localization, specimens reinforced with PVA fiber fail

suddenly due to tensile failure of the fibers. For specimens reinforced with

Spectra fiber, the failure is gradual and controlled by the pulling out of the fibers

accompanied by matrix spalling around the critical crack. Also, for specimens

reinforced with steel fiber, whether Hooked or Twisted, the fibers gradually pull

out up to complete separation; during pull out, twisted fibers untwist leading to

additional matrix cracking. No steel fiber failure was observed for the variables

of this study.

5. For the same volume fraction, and type of steel fiber, the strength of fiber is

important for determining the composite tensile behavior. Specimen reinforced

with high strength steel fiber usually outperform specimen reinforced with regular

strength steel fiber.

6. Changing surface properties of the fiber has important implications. For instance,

specimens reinforced with oiled PVA fibers performed better than specimens

reinforced with non-oiled ones.

7. Variability of properties obtained from direct tensile testing is large and a fact that

cannot be ignored. Both fiber distributions within the specimen and fiber

orientation at any section play a significant role in influencing observed

properties. The more uniform the distribution of fiber is, the less variable the

tensile behavior is.

106

CHAPTER 5

STRESS VERSUS CRACK OPENING DISPLACEMENT TESTS

5.1 Introduction

The post-cracking behavior of high performance fiber reinforced cementitious

composites (HPFRCC) primarily depends on how a typical crack propagates and

responds to stresses. Such information can be clarified by stress versus crack opening

displacement tests (σ-COD). The (σ-COD) test presents vital information on the response

of a crack and allows eventually understanding overall strain behavior of HPFRCC in the

multiple cracking stages, and modeling it. The (σ-COD) test information can be directly

related to the strain at every loading.

Experiments to study (σ-COD) behavior of HPFRCC notched specimens

reinforced with PVA, Spectra, Hooked, and Torex fibers were carried out. Nintety

specimens were tested. The specimens were short Dogbone shaped tensile

specimens,with symmetrical notches at their mid-section. Dimensions are: width = 76.2

mm (3 inches), length = 304.8 mm (12 inches), and depth or thickness = 25.4 mm (1

inch). Additionally, after hardening, notches 12.7 mm (0.5 inch) deep and about 2.54 mm

(0.1 inch) wide were cut at mid-length. The cross-sectional area of a typical specimen, at

the double-notched points, is 1290.32 mm2 (2 in2). The geometric details are shown in

Chapter 3 (Figs. 3.20 and 3.21). The notches were cut using a circular diamond concrete

saw. Four types of fibers, two of which are polymeric (i.e., Spectra and oiled PVA) and

two of which are steel fibers (i.e., High strength Torex fiber and High strength Hooked

107

fiber) are selected in four different volume fractions (i.e., 0.75%, 1.0%, 1.5%, and 2.0%)

to match the direct tension tests described in Chapter 4.

Two crack gauges were placed one on the left and the other on the right, spanning

each notch to measure the displacement (Fig. 5.1). The gauge length was 1 in or 25 mm.

The tests were carried out in the same way as the direct tensile tests, using the same

equipment and the same end grips. Typical test results are presented below.

Figure 5.1: Typical behavior of notch HPFRCCs test

5.1.1 Typical Overall (σ-COD) Response

As the specimen is subjected to the tensile load, the first crack appears, mostly at one

end near or at the notch, at a tensile stress ranging from 180 to 450 psi, (1.241 to 3.102

MPa), depending on the type of fiber and the fiber volume fraction. After initiation the

crack grows in different ways depending on the material (strain hardening, material

transition, or strain softening material). The area where cracking occurred is mostly the

area between the two notches, and can be viewed as a smeared zone of cracking, or

influence zone. When the tensile load and displacement increased, more cracks were

observed as seen in Figs. 5.2a, and 5.2b. Note that often, the first crack did not

necessarily propagate all the way through the section, prior to the formation of other

cracks. In some cases it did, and in others it did not.

108

(a)

(b)

Figure 5.2: Typical behavior of notched specimen, (a) with one crack and (b) with

several cracks

Figure 5.3: Zone of cracking influence in notch specimen

Non-cracked

Non-cracked

Influenced cracking A

109

5.2 Typical Tensile Response (σ-COD) of Double-Notched Specimens

Figure 5.4: Typical crack propagation and localization in HPFRCC

From experimental results, the stress versus crack opening displacement of FRC

composites could be classified into three categories: a) clearly strain hardening material

with multiple cracks, b) material with transition behavior somewhat in between strain-

softening and strain-hardening, and c) clearly strain softening material in which the stress

at cracking is the highest stress. For the materials with transition behavior, two cases are

observed as described below, both having a single crack.

a. Strain Hardening Response. The response of a typical strain-hardening (or

displacement hardening) composite with multiple cracks within the gauge length

is shown in Fig. 5.5a, and a typical photo is shown in Fig. 5.5b. In this example, a

zone of influence around the notched section develops, within which multiple

cracks occur. Eventually, localization failure, similar to that observed in non-

notched tensile specimens, occurs after maximum stress. Both the bond strength

and the length of the fiber (which is of the same order as the gauge length) seem

to influence this behavior. This type of behavior was observed in HPFRCC

reinforced with high strength Hooked fibers (2.0%, 1.5%), Torex fibers (2.0%,

1.5%), and Spectra 2.0%, 1.5%, 1.0%.

110

Figure 5.5 (a): Strain hardening material with

multiple cracks

Figure 5.5(b): Corresponding specimen

b. Transition Behavior. Here two cases are considered. In the first example, the

response of what looks like a strain hardening material with a single major

crack is illustrated in Fig. 5.6a and Fig. 5.6b. After the first cracking point, the

load dropped by about 5% to 30%, then the specimen showed increased load

resistance. The hardening stage was the result of only one single crack. During

the pulling-out process that followed, some additional cracks were seen around

the localization area, while the matrix was spalling out under large

displacement. This type of behavior was observed in HPFRCC reinforced with

Hooked fibers (1.0%, 0.75%), Torex fibers (1.0%, 0.75%), and Spectra fibers

(0.75%).

111

Figure 5.6 (a): Strain hardening material

with single major crack

Figure 5.6 (b): Corresponding specimen

In the second example of transition behavior, a strain softening material

response is observed with a post-cracking stress that first drops after first

cracking, then picks up again but remains smaller than the cracking strength. A

typical response is shown in Fig. 5.7a and a typical photo is shown in Fig. 5.7b.

The specimen’s response is linear elastic up to the first cracking stress. No

multiple cracking is observed. The maximum post-cracking stress is smaller than

the cracking strength but occurs after some displacement or crack opening. After

the maximum post cracking stress, localization is confirmed and the specimen

fails in a tensile softening manner due primarily to fiber pull-out. Only one single

crack was observed in this type of specimen. This type of response was observed

in HPFRCC specimens reinforced with Hooked fibers (0.75%), and PVA fibers

(1.5%, 1.0% and 0.75%).

112

Figure 5.7 (a): Strain softening material

with a post cracking stress that picks up

after a first dip


c. Strain Softening Response. The typical stress versus COD response of a

strain softening composite, where the post-cracking resistance keeps steadily

decreasing after first cracking, is shown in Fig. 5.8a with a corresponding

photo in Fig. 5.8b. The specimen’s behavior is linearly elastic up to the first

cracking point, which also becomes the maximum tensile resistance point, and

the point at which localization starts. After this point, the load drops

significantly. The failure crack opens widely and the load drops either due to

sudden failure of the fibers (PVA), or direct tensile softening manner by partial

fiber pull-out or both. This type of response was observed in FRC specimens

reinforced with PVA fibers (1.5%, 1.0%, and 0.75%)

113

Figure 5.8 (a): Strain softening material


Figure 5.9 provides a comparison of the four different types of observed behavior

using the same scale for stress and crack opening displacement. The magnified scale of

the x axis allows to better see the ascending portion of the stress-COD curve.

.

Figure 5.9: Typical stress-crack opening displacement curves

For comparison purpose, an average curve for each test series or parameter was

determined. The average plot was calculated according to an averaging procedure

114

initially described by Naaman and Najm (1991). The peak load of the average curve is

the average maximum load of the several individual tests, and their displacement is the

average corresponding displacement. The ascending and descending portions of each

individual curve are divided into a chosen number of points (200 points for this research)

and their corresponding loads and averaged. That load is then used as the load of the

average curve for the displacement having the same rank.

Figure 5.10 illustrates the average curve obtained from a series of five specimens

using 2% Spectra fibers by volume. It can be observed that the average curve has a good

similarity with the shape of the individual curves obtained and does provide an intuitively

reasonable average..

Figure 5.10: Stress-displacement curves of notched specimens with 2% Spectra fiber and

average curve

5.3 Notched Mortar Specimens without Fiber

Two notched plain mortar specimens without fibers were tested as control and for

comparison purposes. A typical (σ-COD) response curve is shown in Fig. 5.11.

115

Figure 5.11: Stress-displacement curve of plain mortar matrix without fiber

For specimen-1 and specimen-2 the maximum stress was 238 psi (1.641 MPa)

and 205 psi, (1.413 MPa), respectively, leading to an average of 222.13 psi, 1.531 MPa,

and a displacement at maximum stress of 0.000315 and 0.0002 inch. The energy under

the curves of specimen-1 and specimen-2 is (proportional to) 0.000374 psi and 0.0266 psi

(2.58E-6 to 1.83E-4 MPa). The energy absorption capacity of these specimens is very

low with no ductile behavior.. The non-linear response shown in Fig. 5.11 correspond to

the slow propagation of the crack from one side of the specimen to the other side under

very slow loading rate.

In comparing the maximum stress for the notched series without fiber and the

overall average cracking stress observed from the direct tensile test series (Dogbone

tests), it can be observed that the stress in the notched specimens was about 21.3%

higher. The overall average from the direct tensile tests was 183 psi (1.261 MPa). The

reason could be that in the direct tensile test, the weakest section fails first, while in the

notched tensile test, the section to crack first is pre-selected at the notch location.

116

Table 5.1: Key results for notched specimens without fiber

5.4 Tensile Response (σ-COD) of Double-Notched Specimens with PVA

Fiber

Twelve specimens were tested in this series. Specimen identification and other

details are in Table 5.2. The first letter denotes the type of test (N = Notch tensile test).

The second letter denotes the type of fiber (P = PVA-H fiber). Note that the PVA-H fiber

is the oiled fiber described in Chapter 4. The third letter denotes the volume fraction of

fiber, (2.0%, 1.5%, 1.0%, and 0.75%). Each series used 3 specimens. The stress-

displacement curves are described in Figs. 5.12-5.15, and discussed next.

Table 5.2: Test series of notched specimens reinforced with PVA fiber

Identification

of Specimen

Type of

Specimen

Number of

Specimens

Type of

Matrix

Type of

Fibers

Volume

of

Fibers

(Vf(%)

N-P-2 Notch 3 Mortar Oiled-PVA 2.00%

N-P-1.5 Notch 3 Mortar Oiled-PVA 1.50%

N-P-1 Notch 3 Mortar Oiled-PVA 1.00%

N-P-0.75 Notch 3 Mortar Oiled-PVA 0.75%

117

Figure 5.12 (a): (σ-COD) curves of

specimens reinforced with 0.75% PVA

fiber

Figure 5.12 (b): Photo of tested specimens

reinforced with 0.75% PVA fiber


specimens reinforced with 1% PVA fiber


reinforced with 1% PVA fiber

118


specimens reinforced with 1.5% PVA fiber


reinforced with 1.5% PVA fiber Notch

FRC-PVA 1.5%


specimens reinforced with 2% PVA fiber


reinforced with 2% PVA fiber

119

Figure 5.16: Comparison of average (σ-COD) curves with PVA fiber

Testing results show an elastic response up to the first cracking, with an average

cracking stress of 314.57 psi (2.168 MPa). However, the response was rather brittle. The

maximum crack opening displacement was small (i.e., approximately ten times smaller

than observed from similar specimens reinforced with Spectra, Hooked, and Torex fiber).

Although strain hardening behavior is evident (i.e., the stress increases after the specimen

cracks), there is no observable multiple cracking behavior. The strain hardening and

nonlinearity may have resulted from the slow propagation of the crack from one side to

the other side of the specimen. Because of the test procedure where the end grips are

hinged, any crack could create some bending leading to a non linear response. Regarding

the effectiveness of fiber, the highest improvement in tensile resistance is found at a fiber

volume fraction of 1.5%, with an average maximum tensile resistance of 566.52 psi

(3.906 MPa). Ductility and energy absorption capacity were low in comparison to the

other fibers tested in this study.

120

(a)

(b)

Figure 5.17: Definition of localization start point, (a) for specimen with only one

global maximum stress and (b) for specimen with a second local maximum

One interesting phenomenon specific to the test series with PVA fiber is related to

the localization after the peak stress. After the first crack, some FRC specimens

reinforced with PVA fiber exhibit displacement hardening behavior with increasing load.

However, the localization phase may not start immediately after the peak stress, (Fig

5.17a). Instead, the load drops and pick-up again, reaching a second local maximum post-

cracking stress, which is lower than the first one. Localization starts after this second

maximum as shown in Fig. 5.17b, and is followed by tensile softening behavior.

121

Table 5.3: Summary of test results for notched specimens reinforced with PVA fiber

(US-units)

Table 5.4: Summary of test results for notched specimens reinforced with PVA fiber

(SI-units)

122

Figure 5.18: First cracking stress

Figure 5.19: Displacement at first cracking

stress

Figure 5.20: Energy at first cracking stress

A summary of tests results for all notched specimens reinforced with PVA fiber is

given in Tables 5.3 and 5.4. The stress at first cracking, the corresponding displacement

and energy are plotted in Figs. 5.18 to 5.20 versus the volume fraction of fiber. It can be

observed that both the cracking stress and corresponding displacement vary little with Vf.

The first cracking stress is on average about 315 psi (2.17 MPa) (Fig. 5.18). Here it is

smaller than that observed in the direct tensile test specimens which were around 434 psi

(2.99 MPa).

123

Figure 5.21: Maximum stress

Figure 5.22: Displacement at maximum

stress

Figure 5.23: Energy at maximum stress

Figures 5.21 to 5.23 describe the variation of maximum stress, the corresponding

displacement, and the energy versus volume fraction of fiber. No particular trend could

be detected. It is sufficient to say that the maximum stress ranged from 377 to 566 psi

(2.599 to 3,902 MPa) with the maximum occurring at 1.5% fiber content by volume, and

the displacement was smaller than 0.16 mm in all cases. The energy at maximum stress

ranged from about 0.5 to 2 psi-in. (0.088 to 0.35 MPa-mm) (Fig. 5.23)

124

Figure 5.24: Displacement at localization

start

Figure 5.25: Energy at localization start

The observed displacement at localization start ranged from 0.0028 to 0.00625

inch (0.0713 to 0.159 mm), with an average of about 0.00423 inch (0.00107 mm) (Fig.

5.24). The corresponding energy at localization start ranged from 0.874 to 2.16 psi-in.

(0.153 to 0.378 MPa-mm) with an average of 1.447 psi-in (0.253 MPa-mm), as shown in

Fig. 5.25.

Although specimens reinforced with PVA had several characteristics different

from those with the other fibers tested in this study, there seems to be some correlation

between the (σ-COD) test results and the test results observed from the direct tensile test

series described in Chapter 4, particularly the failure mode and the lack of multiple

cracking.

Concluding Remarks (PVA Fiber)

1. No multiple cracking was observed near the main crack in the notched tensile

specimens at all volume fractions tested. However, strain hardening behavior was

observed in specimens with high volume fractions of PVA fiber. Such behavior

could be attributed to the slow propagation of a single crack from one end of the

section to its other end.

2. Only single cracks were observed in these tests. Each crack was fine and well

defined with no large damage zone around it .

125

3. For the given parameters of this study, the optimum volume fraction of fiber is

1.5%, the same as observed from the direct tensile tests (Chapter 4). However, the

average maximum stress of the notched prisms (about 3.9 MPa) was slightly

higher than that obtained from the dogbone tensile prisms (about 3.7 MPa).

4. Comparing the displacement at maximum stress between specimens with high

strength Hooked, Torex, Spectra, and PVA fiber,the smallest displacement was by

far with the PVA fiber. Typically such displacement was 10 times smaller than

observed from specimens with the other fibers.

5.5 Tensile Response (σ-COD) of Double-Notched Specimens with

Spectra Fiber

Fifteen notched tensile specimens reinforced with Spectra fiber were tested.

Specimen identification is described in Table 5.5. The first letter denotes the type of test

(N = Notch tensile test, stress-crack opening displacement test). The second letter denotes

the type of fiber (S = Spectra fiber) and the third letter denotes the volume fraction of

fiber, (2.0%, 1.5%, 1.0%, or 0.75%). Due to insufficient strength at the anchorage zone,

some specimens with Vf = 1.5% were reinforced with carbon mesh externally in the

anchorage zone which was bonded with epoxy. Each test series had two to five

specimens. The stress versus crack opening displacement results are described in Figs.

5.26 to 5.29 for each volume fraction of fiber and a comparative summary of average

curves is given in Fig. 5.30.

Table 5.5: Test series of notched specimens reinforced with Spectra fiber

Identification

of Specimen

Type of

Specimen

Number of

Specimen

Type of

Matrix

Type of

Fibers

Volume

of

Fibers

Vf(%)

N-S-7.5 Notch 2 Mortar Spectra 2.00%

N-S-1 Notch 3 Mortar Spectra 1.50%

N-S-1.5 Notch 5 Mortar Spectra 1.00%

N-S-2 Notch 5 Mortar Spectra 0.75%

126


specimens reinforced with 0.75% Spectra


reinforced with 0.75% Spectra





127









128

Figure 5.30: Comparison of average (σ-COD) curves with Spectra fiber

Overall the specimens showed ductile behavior following an initial elastic

response. The average stress at first cracking was about 200 psi (1.38 MPa). The

maximum tensile resistance was 419 psi, or 2.89 MPa at a fiber volume fraction of 1.5%.

This stress level is comparable to the maximum stress observed with the direct tensile

tests using Dogbone specimens (Chapter 4). An increase in volume fraction of fiber leads

to an increase in ductility, energy absorption capacity and additional multiple cracks in

the zone of influence around the notched section. Multiple cracking was observed even

for some specimens with the lowest fiber content of 0.75% by volume. At the higher

volume fraction of fiber, the cracking zone extended a distance about the same as the

length of the fiber.

129



stress


Note that the first cracking stress of 200 psi, (1.38 MPa) was slightly lower than

that of the notched specimens without fiber, (Fig. 5.11 and Fig. 5.31). This may be the

result of a large amount air bubbles entrapped within a specimen due to mixing

difficulties with long fibers, resulting in non-uniform specimen composition.

130



stress


Figures 5.33 to 5.35 show that an increase in volume fraction of fibers leads to an

increase in maximum stress, displacement at maximum stress, and related energy

absorption capacity, up to a volume fraction of fiber of 1.5%, after which the trend tapers

off. The maximum stress for Vf = 1.5% awas 566 psi (3.9MPa). When the volume

fraction of fiber increases from 1.5% to 2%, the energy at maximum stress drops by

about 11%.

131

Figure 5.37: Typical multiple cracking in notched specimens with Spectra fiber

The extent of multiple cracking was significant with the use of Spectra fibers (Fig.

5.37). This led to a large displacement at maximum stress and prior to localization.

Several peaks could be observed making it difficult to select which one was the precursor

of localization.

Figure 5.38: Displacement at the end of

multiple cracking stage


cracking stage

At the end of the multiple cracking stage and onset of localization, the

displacement and corresponding energy increase almost linearly with the volume fraction

of fibers (Figs. 5.38 and 5.30). These can be represented by the following linear

equations:

D = 0.0751 Vf – 0.0237 (4.1)

E = 37.266Vf -19.981 (4.2)

where D represents displacement, and E represents energy.

132

Table 5.6: Summary of test results for notched specimens reinforced with Spectra fiber

(US-units)

133

Table 5.7: Summary of test results for notched specimens reinforced with Spectra fiber

(SI-units)

Concluding Remarks (Spectra Fiber)

1. Multiple cracking behaviors around the notched section (equivalent to smeared

cracking) were clearly observed. Such behavior encourages strain hardening (or

high performance) response in direct tension as indeed observed in the direct

tensile tests.

2. The cracking zone on either side of the notch area (area between two notches) is

significant, containing several cracks. Its extent increased with an increase in

volume fraction of fibers. Specimens containing high volume fraction of fibers

usually outperformed those containing low volume fraction of fibers.

3. No consistent correlation could be established between the fiber content and the

first cracking stress, the strain at first cracking stress, and the energy at first

cracking stress. These variables seemed independent of the volume fraction of

fibers for the range of fiber content tested. However, the maximum stress, strain

134

at maximum stress, and energy at the end of multiple cracking stage all increased

with an increase in volume fraction of fiber.

4. Keeping in mind mixing difficulty and air entrapment, the best performance was

obtained at a volume fraction of 1.5% (not 2%) and is consistent with the results

from the direct tensile (dogbone) tests.

5.6 Tensile Response (σ-COD) of Double-Notched Specimens with High

Strength Hooked Fiber

Twenty-seven notched specimens reinforced with Hooked fibers were tested. The

test series are denoted as series N-H, having four volume fractions of fiber selected

(0.75%, 1.0%, 1.5%, and 2.0%). They are described in Table 5.8. In order to better

evaluate variability of the results, the number of specimens tested at fiber volume

fractions of 1.5% and 2.0%, was 12 and 9, respectively.

Figures 5.41 to 5.44 give the stress versus crack opening displacement curves at

the four fiber volume fractions tested. Figure 5.45 provides a comparison of the average

curves obtained for each volume fraction.

Table 5.8 Test series of notched specimens reinforced with Hooked fiber

Identification

of Specimen

Type of

Specimen

Number of

Specimens

Type of

Matrix

Type of

Fibers

Volume

of

Fibers

(Vf) (%)

N-H-2 Notch 9 Mortar Hooked 2.00%

N-H-1.5 Notch 12 Mortar Hooked 1.50%

N-H-1 Notch 3 Mortar Hooked 1.00%

N-H-0.75 Notch 3 Mortar Hooked 0.75%

135


specimens reinforced with 0.75% Hooked

fiber


reinforced with 0.75% Hooked fiber


specimens reinforced with 1.0% Hooked

fiber

Figure 5.41(b): Photo of tested specimens


136

Figure 5.42: (σ-COD) curves of specimens


Figure 5.43: (σ-COD) curves of specimens


Figure 5.44: Comparison of average (σ-COD) curves with Hooked fiber

Testing results, especially Figs. 5.42 and 5.43, revealed large variations in

response for the same volume fraction of fibers. For example, in Fig. 5.42, for specimens

reinforced with 1.5% Hooked fiber by volume, the maximum stress ranged from 560 psi

to 1193 psi. (3.86 to 8.23 MPa). The overall specimens’ behavior was relatively ductile.

An elastic response was observed up to the first cracking with an average cracking stress

of 332 psi (2.289 MPa). However, on average, the higher the volume fraction of fiber the

higher the stress at first cracking and the maximum stress (Fig. 5.44).

137

Figures 5.45 to 5.47 describe the variation of the peak point properties with the

volume fraction of fiber. It can be observed that from Vf = 0.75% to Vf = 1.5% there is

an increase in the the first cracking stress, the displacement at first cracking stress, and

the corresponding energy. However, these taper off between 1.5% and 2.0% volume

fractions of fiber. Also the damage in the failure zone around the notched section

increases when increasing the volume fraction of fiber.

When comparing variability, it seems from the limited tests that the greater the

volume fraction of fiber the greater the variability. Indeed specimens containing 2.0%

volume fraction of fiber have a standard deviation of 294 psi (2.03 MPa) for the

maximum stress versus 231 psi, (1.59 MPa ) for the specimens with 1.5% fiber volume

fraction. More tests are needed to ascertain this result.



stress


138



stress


Figures 5.48 to 5.50 relate to the properties of the peak point. The maximum

stress clearly increases when the volume fraction of fiber increases and tapers off

between 1.5% and 2%. However, the displacement at maximum stress does not follow a

consistent trend but can be assumed to slightly decrease, and the energy absorption

capacity remains almost constant.

139




cracking stage

The displacement at the end of the multiple cracking and the corresponding

energy appear not to be affected by the volume fraction of fiber, as illustrated in Figs.

5.51 and 5.52. Additional tests may be needed to better understand and ascertain this

observation.

140

Table 5.9: Summary of test results for notched specimens reinforced with Hooked fiber

(US-units)

141

Table 5.10: Summary of test results for notched specimens reinforced with Hooked fiber

(SI-units)

Concluding Remarks (Hooked Fiber)

1. Numerous multiple cracks clearly occurred around the notched section.

Displacement (strain) hardening behavior was observed within the 1 in (25 mm)

guage length. The higher the volume fraction of fibers the more extended was

the cracked region. Specimens with 2.0% and 1.5% volume fractions were

observed as having a larger number of cracks. (about 5 equivalent cracks)

2. Generally the higher the volume fraction of fibers the better the post-cracking

strength. However, the displacement at maximum stress and the energy at

maximum stress were almost independent of the fiber volume fraction.

142

3. The stress at first cracking, the corresponding displacement and related energy

increased slightly with the volume fraction of fibers. A high variability in test

results was clearly observed and will be further addressed (Chapter 7).

5.7 Tensile Response (σ-COD) of Double-Notched Specimens with Torex

Fiber

Seventeen notched sepciemens reinforced with Torex fiber were tested. The test

series are identified as series N-T; four volume fractions of fiber were used (0.75%,

1.0%, 1.5%, and 2.0%). Table 5.11 describes the series identification and the number of

specimens tested for each series. The first letter denotes the type of test (N for notch

tensile test/stress–crack opening displacement test). The second letter denotes the type of

fiber (T = Torex fiber). The third letter denotes the volume fraction of fiber, (2.0%, 1.5%,

1.0%, or 0.75%). The Torex fiber was high strength steel as described in Chapter 4.

The stress versus crack opening displacement curves are plotted in Figs. 5.54 to 5.57 and

a comparison of average curves are given in Fig. 5.58. Test results are summarized in

Tables 5.12 and 5.13.

Table 5.11: Test series of notched specimens reinforced with Torex fiber

Identification

of Specimen

Type of

Specimen

Number of

Specimens

Type

of

Matrix

Type

of

Fibers

Volume

of

Fibers

(Vf(%)

N-T-2 Notch 6 Mortar Torex 2.00%

N-T-1.5 Notch 5 Mortar Torex 1.50%

N-T-1 Notch 3 Mortar Torex 1.00%

N-T-0.75 Notch 3 Mortar Torex 0.75%

143


specimens reinforced with 0.75% Torex fiber

Figure 5.53 (b): Photo of tested

specimens reinforced with 0.75% Torex

fiber





fiber

144



Figure 5.55(b): Photo of tested


fiber





fiber

145

Figure 5.57: Comparison of average (σ-COD) curves with Torex fiber

Overall the response of the notched specimens with Torex fibers was excellent.

As the load increased, the displacement increased and some multiple fine cracking

developed in the zone around the notched section. Displacement hardening (or strain

hardening) occurred even at the low fiber volume content of 0.75%. For the fiber volume

content of 0.75% and 1%, the results within series were consistent and with low

variability (Figs. 5.53 and 5.54). A higher variability was observed with the series at

1.5% and 2%. This was for two reasons: first, at the higher fiber volume content, some

critical cracks occurred outside the notched section (Fig. 5.56b), and second, due to the

high tensile stresses failure occurred at the gripps in preliminary tests, so the specimen

had to be reinforced at the gripps with epoxy bonded carbon mesh. Overall the higher the

fiber volume fraction, the better the response is. While the maximum tensile stress

observed was significantly higher than with Spectra fibers, the zone of influence for

multiple cracks around the notched section was small and extended only about half an

inch (13 mm). Maximum tensile stresses were higher than for the series of tests using

dogbone specimens (Chapter 4).

All curves showed an initial elastic response up to the first cracking and the

average cracking stress was about 400 psi (2.75 MPa). Figures 5.58 to 5.60 describes the

146

variation of first cracking stress, displacement at first cracking and corresponding energy

with the volume fraction of fiber. It can be observed that while the cracking stress

remains almost a constant, both the displacement and the corresponding energy increase

with the volume fraction of fiber.

Figure 5.58: First cracking stress Figure 5.59: Displacement at first cracking

stress


Figures 5.61 to 5.63 describe the variation of the properties at peak point of the

curve (maximum stress) with the volume fraction of fiber. The maximum stress increased

from 757 psi to 1176 psi (5.22 to 8.11 MPa) when the volume fraction increased from

0.75% to 2%. The corresponding displacement and energy also increased up to 1.5%

fiber volume content then tapered off. Best results were obtained at 1.5% fiber content

147

and may due to the fact that it is more difficult to mix 2% fibers by volume leading to

increased air entrapment.



stress


148




cracking stage

Since some additional micro cracks occurred after the peak stress point, a point

indicating the end of multiple cracking was defined. Figures 5.64 and 5.65 describe the

variation of displacement at the end of the multiple cracking stages and the corresponding

energy. Both increase with the fiber volume fraction and taper off at higher values.

In comparing the behavior of notched specimens using Torex fibers with that of

similar specimens using either Hooked or Spectra of PVA fibers, one can state that, while

Torex fiber led to the best overall performance in terms of strength and energy, Spectra

fiber led to the best ductility although the specimen’s strength was half that achieved with

the Torex fiber.

149

Table 5.12: Summary of test results for notched specimens reinforced with Torex fiber

(US-units)

150

Table 5.13: Summary of test results for notched specimens reinforced with Torex fiber

(SI-units)

Concluding Remarks (Torex Fiber)

1. The use of Torex fibers led to significant multiple cracking around the notched

section, and displacement (or strain) hardening behavior was observed.

Significant ductility and energy absorption were also observed prior to

localization. Multiple cracking was observed around the notched section, even at

0.75% fiber content.

2. The higher the volume fraction of fibers, the higher the maximum post-cracking

stress and related energy absorption capacity, and the larger the number of

equivalent cracks. Everything else being equal, maximum stresses here exceeded

those obtained from direct tensile tests.

3. The stress at first cracking did not seem to depend much on the fiber volume

fraction, while the displacement at first cracking increased with Vf.

151

4. The damaged area around the notched section was usually smaller than that using

Spectra fibers. This could be attributed to the fact that the Spectra fibers used

were 38 mm in length compared to the 30 mm Torex fibers.

5. The variability in test results obtained with Torex fibers was smaller than that

with Spectra and Hooked fibers.

5.8 Comparison of Test Series Reinforced with Different Fibers

(a)

(b)

(c)

(d)

Figure 5.66: Comparison of stress-crack opening displacement for the four fibers

used at (a) Vf=0.75%, (b) Vf=1.0%, (c) Vf=1.5%, and (d) Vf=2.0%,

152

Figures 5.66a to 5.66d show average stress versus COD curves for the four fibers

tested, each at four different volume fractions. The scales for each fiber are different in

order to better clarify the ascending portion of each curve, or to accommodate the

increase in maximum stress. In every case, it can generally be said that increasing the

fiber volume fraction improves the overall response (σ-COD) of the composite, not only

in terms of stress or resistance, but also in terms of ductility and energy absorption

capacity. This trend may taper off at higher volume fractions; this may be due to the

increasing difficulty to mix fibers in larger volumes and the increased porosity of the

mix, which can nullify the potential benefits of additional fibers. It may also be due to

group effect, where the efficiency of one fiber is reduced when a large number of fibers is

pulling out simultaneously from the same area. In the examples of this study, the

difference in stress-COD between using 1.5% and 2% fibers by volume may not justify

the increase in cost due to the fibers added. In comparing the effectiveness of different

fibers, it can be said that consistently the Torex fiber led to better overall performance

than the other fibers tested. It was followed by high strength Hooked, Spectra then PVA

fiber. At 1% fiber content, the maximum stress achieved with the Torex fiber is almost

40% higher than that with Hooked fiber, and more than twice that with Spectra and PVA

fibers. At 2% fiber content, the maximum stress achieved with Torex fiber is 20%

higher than that with the Hooked fiber, but almost three times that achieved with the

Spectra and PVA fiber.


The following conclusions can be drawn from this study:

1. The stress versus crack opening displacement (COD) of FRC composites can be

classified into four types: clearly displacement (or strain) hardening with multiple

cracks; clearly displacement (or strain) softening with a single localized crack ;

and two cases in between, a displacement (or strain) hardening material with a

single major crack ; a displacement (or strain) softening material with post

cracking strength, able to pick up almost up to the cracking stress.

153

2. For the same volume fraction of fibers, specimens with Torex and High strength

Hooked fibers showed better overall behavior than specimens with Spectra and

PVA fibers. Moreover, their post-cracking strength was 1.5 to 3 times the strength

at first cracking, while for specimens with PVA fibers, the post-cracking strength

was only 10% to 50% higher than the cracking strength.

3. For the same type of fiber, increasing the volume fraction leads to a marked

improvement in post-cracking strength, ductility, and energy absorption capacity

of the composites.

4. The crack opening in specimens with Torex, high strength Hooked and Spectra

are the result of fiber-pulling out. However, the damage in specimens with PVA

fibers was due to fiber breaking.

In this study only one mortar matrix composition and strength was used. It is

likely that the results would be different with different strength matrices. However, it

also likely that the typical observations for the shape of the stress versus COD curves

will be similar to what is observed in this study.

154

CHAPTER 6

PROPOSED MODEL FOR POST-CRACKING BEHAVIOR OF HPFRCC UNDER TENSION

6.1 Introduction

Figure 6.1: Proposed diagram of HPFRCC model

It is generally believed that from the micromechanical properties, it is possible to

derive a model of a single fiber bridging a crack and use that to predict the overall

behavior of the composite at the material’s level and even the structure. However, the

result of a single-fiber is not sufficient to design a fiber-reinforced structural element. The

main reason is, when the macro-mechanical properties are investigated, the randomly

oriented discontinuous fibers introduce additional parameters, such as the uncertainty of

the number of fibers bridging the crack, the pulley effect, and matrix spilling, which

155

considerably affect the overall macro-mechanical behavior. Therefore, a more realistic

model in tension is required to account for these effects.

Generally, the stress-strain response of HPFRCCs in tension exhibits linearly

elastic behavior up to the first cracking. Beyond the elastic limit (first structural

cracking), the crack is assumed to extend over the composite cross-section. Hence, the

crack is resisted only by the bridging fibers. The first crack strength is defined as the

applied tensile stress at which the crack percolates throughout a cross-section. After first

cracking, the multiple cracking stage takes place, and the tensile member can be viewed

as divided into numerous segments in series, like a chain (Fig. 6.2). The width of each

crack depends on the stress transferred from the fibers to the matrix. After the multiple

cracking processes are completed, no further cracks form. Then, under further loading or

straining, localization takes place as a result of the opening of the main critical crack and

failure follows by reduction of the tensile resistance.

Figure 6.2: Typical tensile response of HPFRCC and chain model

In this Chapter, a model to predict the post-cracking stress-strain response of

HPFRCC in tension is explained. Details are given in seven sections. First cracking

stress, Strain at first cracking stress, and maximum post-cracking stress or Ultimate

cracking strength are explained in Sections 6.2, 6.3 and 6.4. Multiple cracking behavior

and strain at maximum stress or at Ultimate are discussed in Section 6.5. Localization of

the critical crack and the descending branch of the curve (Fig. 6.2) are discussed in

Section 6.6. The correlation between direct tensile tests and stress versus crack opening

156

displacement tests is addressed in Section 6.7. A flowchart of the proposed stress-strain

model and the computational steps are demonstrated in Section 6.8. Finally, a preliminary

verification and calculation to illustrate the application of the proposed model is

presented in Section 6.9.

6.2 First Cracking Stress

An equation to predict the first cracking stress is obtained from earlier research

conducted by Naaman (1972, 1974, and 1987) as

σcc = σmu (1-Vf) + α1 α2 τ Vf L / d (6.1)

where Vf = Volume fraction of fiber

L = Fiber length

d = Fiber diameter

L/d = Fiber aspect ratio

σmu = Tensile strength of the matrix

τ = Average bond strength at the fiber matrix interface

α1 = Coefficient to describe the fraction of bond mobilized at first crack

α2 = Coefficient of fiber orientation in the uncracked state

The product of coefficients α1 α2 was obtained from the experimental data on the

direct tensile tests described in Chapter 4. Details are given next. The following fiber

properties were used for all calculations in this chapter:

PVA Fiber (oiled as described in Chapter 4)

Diameter = 0.19 mm = 7.48x10-3 inch

Length = 12 mm = 0.5 inch

Equivalent bond strength τ = 3.5 Mpa = 510 psi (Guerrero, 1999)

l/d = 67

157

Spectra Fiber



Equivalent bond strength τ = 0.62 Mpa = 90 psi (Li and Lieung, 1991)

l/d = 1000

Hooked Fiber (high strength as described in Chapter 4)



Equivalent bond strength τ = 5.1 Mpa = 740 psi (Guerrero, 1998)

l/d = 75

Torex Fiber (high strength as described in Chapter 4)



Equivalent bond strength τ = 6.8 Mpa = 992 psi (Sujivorakul, 2002)

l/d = 100

6.2.1 FRC Reinforced with PVA Fiber

The average first cracking stress of direct tensile test series reinforced with PVA

fiber is given in Table 6.1 for each volume fraction of fiber tested, and plotted in Fig. 6.3.

The following values of variables were applied to Eq. (6.1): the tensile strength of the

matrix is taken equal to 182 psi (1.252 MPa) as obtained from tests in Chapter 4; L/d =

67; the bond strength is taken equal to 510 psi (3.52 MPa) as suggested in prior tests by

Guerrero, 1998; The product (α1 α2) can then be calculated from Eq. (6.1) for each fiber

volume fraction. The results are plotted in Fig. 6.4. While the product varies from about

0.45 to 0.67, no clear trend could be detected. Thus an average value is suggested over

the range of fiber volume fractions tested, and corresponds to α1 α2 = 0.5204 (Fig 6.4).

158

Table 6.1: Average first cracking stress of tensile test series with PVA fiber

Figure 6.3: Variation of first cracking stress

for test series with PVA fiber

Figure 6.4: Product α1α2 versus Vf

6.2.2 HPFRCC Reinforced with Spectra Fiber

The average first cracking stress of direct tensile test series reinforced with

Spectra fiber is given in Table 6.2 for each volume fraction of fiber tested, and plotted in

Fig. 6.5. The following values of variables were applied to Eq. (6.1): the tensile

strength of the matrix is taken equal to 182psi (1.252 MPa); L/d = 1000; the bond

strength is taken equal to 89.923 psi (0.62 MPa) as suggested in prior tests (Li and

Lieung, 1991). The product (α1 α2) can then be calculated from Eq. (6.1) for each fiber

volume fraction. The results are plotted in Fig. 6.6. The corresponding value for Vf =

0.75% was disgarded since it is too much out of range. The remaining values α1 α2 for Vf

ranging from 1% to 2% led to an average product α1 α2 = 0.0424.

159

Table 6.2: Average first cracking stress of tensile test series with Spectra fiber

Figure 6.5: Variation of first cracking stress

for test series with Spectra fiber Figure 6.6: Product α1α2 versus Vf

6.2.3 HPFRCC Reinforced with Hooked Fiber

The average first cracking stress of direct tensile test series reinforced with

Hooked fiber is given in Table 6.3 for each volume fraction of fiber tested, and plotted in

Fig. 6.7. The following values of variables were applied to Eq. (6.1): the tensile strength

of the matrix is taken equal to 182 psi (1.252 MPa); L/d = 75; the bond strength is taken

equal to 740 psi (5.1 MPa) as suggested in prior tests (Guerrero, 1998). The product (α1

α2) can then be calculated from Eq. (6.1) for each fiber volume fraction. The results are

plotted in Fig. 6.8. The corresponding value for Vf = 0.75% was discarded since it is too

much out of range. The remaining values of α1 α2 for Vf ranging from 1% to 2% led to an

average product α1 α2 = 0.295.

160

Table 6.3: Average first cracking stress of tensile test series with Hooked fiber

Figure 6.7: Variation of first cracking stress for

test series with Hooked fiber


6.2.4 HPFRCC Reinforced with Torex Fiber

The average first cracking stress of direct tensile test series reinforced with Torex

fiber is given in Table 6.4 for each volume fraction of fiber tested, and plotted in Fig. 6.9.

The following values of variables were applied to Eq. (6.1): the tensile strength of the

matrix is taken equal to 182 psi (1.252 MPa); L/d = 100; the bond strength is taken

equal to 992.28 psi (6.8 MPa) as suggested in prior tests [Sujivorakul, 2002]. The

product (α1 α2) can then be calculated from Eq. (6.1) for each fiber volume fraction. The

results are plotted in Fig. 6.10. The product α1 α2 remained almost constant for Vf ranging

from 0.75% to 2% with an average α1 α2 = 0.094.

161

Table 6.4: Average first cracking stress of tensile test series with Torex fiber

Figure 6.9:Variation of first cracking stress

for test series with Torex fiber


6.3 Strain at First Cracking Stress

The average strain at first cracking stress of HPFRCC specimens reinforced with

PVA fiber is around 0.000172, which is 3 times smaller than observed from the

cementitious material without fiber (0.0005). Clearly the presence of fiber in the

composite increases the strain at first cracking stress. This may also be due to the slow

growth of the crack from one side of the specimen to the other side leading to

nonlinearity and strain increment.

Also, the average first cracking strain of HPFRCC specimens reinforced with

Spectra fiber is 0.000985, twice as high as that observed from the cementitious material

without fiber (0.0005). The increase in strain at first cracking may also be explained in a

162

similar manner as for PVA fiber.

The average strain at first cracking stress of HPFRCC specimens reinforced with

Hooked fiber is 0.000287. This number is lower than observed from the control

specimens without reinforcement (0.000517 by almost 45%).

The average strain at first cracking stress of HPFRCC reinforced with Torex fiber

specimens is 0.000224. This number is lower than expected from specimens without

reinforcement (0.000517) by 43% and similar to the strain observed with Hooked fiber.

6.4 Maximum Post-Cracking Stress or Ultimate Stress

The ultimate strength or maximum post-cracking stress (σpc) can be modeled by

the following equation [Naaman, 1972, 1987, 2003]:

σpc = λpc τ Vf L/d (6.2)

where:

λpc = λ1 λ2 λ3

λ1 = Expected pull out length ratio

λ2 = Efficiency factor of orientation in the cracked state

λ3 =Group reduction factor associated with the number of fibers pulling-out per

unit area (or density of fiber crossing the composites)

λ2 = 4 α2 λ4 λ5

λ4 =Pulley effect; Fiber pulls at an angle creating a pulley effect for flexural fibers

plastification or matrix damage for each fiber

λ5 = General reduction in pull-out response when fibers are oriented at greater

than 60o; they become ineffective due to damage around crack and matrix

spelling

In this study, it was decided to simply focus on the global coefficient λpc and estimated it

for the test results.

163


The average value of maximum post-cracking stress (or ultimate strength) of direct

tensile test series reinforced with PVA fiber is given in Table 6.5 for each volume

fraction of fiber tested, and plotted in Fig. 6.11. The following values of variables were

applied to Eq. (6.2): L/d = 67; bond strength = 510 psi (3.52 MPa) as described in

Section 6.2. The coefficient λpc can then be calculated from Eq. (6.2) for each fiber

volume fraction. The results are plotted in Fig. 6.12. An almost linear response is

observed. The corresponding line is given by the following equation: λpc = - 0.707 Vf +

2.093. Such a response is not surprising since group effect is important and leads to a

reduction in fiber efficiency when the number of fibers increases.

Table 6.5: Average post-cracking strength of tensile test series with PVA fiber

Figure 6.11: Variation of maximum stress

for test series with PVA fiber

Figure 6.12: Cefficient λpc versus Vf

164



tensile test series reinforced with Spectra fiber is given in Table 6.6 for each volume


applied to Eq. (6.2): L/d = 1000; bond strength = 89.923 psi (0.62 MPa) as described in


volume fraction. The results are plotted in Fig. 6.14. An almost linear response is

observed similarly to the case with PVA fiber, and can be explained by the reduction in

fiber efficiency when the number of fibers increases.

Table 6.6: Average post-cracking strength of tensile test series with Spectra fiber


for test series with Spectra fiber

Figure 6.14: Coefficient λpc versus Vf

165

The linear relation described in Fig. 6.14 is given by:

λpc = -20.8Vf + 0.63 (6.3)



tensile test series reinforced with Hooked fiber is given in Table 6.7 for each volume


applied to Eq. (6.2): L/d = 30; bond strength = 740 psi (5.1 MPa) as described in Section

6.2. The coefficient λpc can then be calculated from Eq. (6.2) for each fiber volume

fraction. The results are plotted in Fig. 6.16. Ignoring the value at Vf = 0.75%, an almost

linear response is observed similarly to the case with PVA and Spectra fiber, and can be

explained by the reduction in fiber efficiency when the number of fibers increases.

Table 6.7: Average post-cracking strength of tensile test series with Hooked fiber

166


for test series with Hooked fiber



λpc = -55.6Vf + 2.58 (6.4)



tensile test series reinforced with Torex fiber is given in Table 6.8 for each volume


applied to Eq. (6.2): L/d = 100; bond strength =992.28 psi (6.8 MPa) as described in


volume fraction. The results are plotted in Fig. 6.18. Here also, an almost linear

response is observed similarly to the case with PVA, Spectra, and Hooked fibers, and can

be explained by the reduction in fiber efficiency when the number of fibers increases.

167

Table 6.8: Average post-cracking strength of tensile test series with Torex fiber

Figure 6.17: Variation of maximum stress for

test series with Torex fiber



λpc = -26.6Vf + 0.97 (6.5)

6.4.5 Comparison of Coefficient λpc for Different Fibers

The linear relationships related the coefficient λpc to the volume fraction of fiber

are plotted in Fig. 6.19 for the different fibers tested.

168

Figure 6.19: λpc of HPFRCC and Vf

6.5 Multiple Cracking Behavior and Strain at Maximum Stress

Generally, the number of cracks in strain-hardening FRC composites increases

with the volume fraction of fiber, up to a value corresponding to crack saturation. Crack

saturation may occur prior to the maximum stress, and in some instances was observed to

continue after the maximum stress. The variation of average crack spacing and widths

are described in the following sections for the tensile test series with Hooked, Spectra and

Torex fiber. No multiple cracking was observed in this study (Chapter 4) for the test

series with PVA fiber and thus they are not discussed further.

169

6.5.1 Test Series with Hooked Fiber

Figure 6.20: Typical increase in number of cracks with fiber volume fraction

Figure 20 illustrates the increase in number of cracks at saturation for the test

series described in Chapter 4 for Hooked fiber. The extension to Sifcon is taken from

another investigation [See appendix]. Figure 6.21 shows photographs of a typical crack

development with increasing load for a test specimen with Hooked fiber.

Figure 6.21: Crack formation under increasing load in test series reinforced with 1.5%

Hooked steel fiber

170

For a given cracking condition, an equivalent number of cracks can be

determined. It is defined as the total length of cracks observed at a given load, divided by

the specimen’s width. Once an equivalent number of cracks is obtained, an average crack

spacing can be calculated by dividing the gauge length by the equivalent number of

cracks. The largest number of cracks, thus the smallest average crack spacing, occur at

crack saturation yielding a useful number that can be later related to average crack width.

Such calculations were carried out for the direct tensile test series described in Chapter 4.

Results related to each type of fiber are described next.

Figures 6.22 and 6.23 illustrate the variation, respectively, of average crack

spacing and average crack width at saturation versus the volume fraction of Hooked fiber.

Note that at a volume fraction of 0.75%, the composite may have been in a transition

between strain-softening and strain hardening, showing only a couple of equivalent

cracks. If this data point is ignored, then linear relationships could be developed between

crack spacing and width versus the fiber volume fraction.

Figure 6.22: Average crack spacing at

saturation for test series with Hooked fiber

Figure 6.23: Average crack width at

saturation for test series with Hooked fiber

The relationship obtained for the average crack spacing with Hooked fiber is given by:

Xd = -30.4 Vf + 1.266 (in) (6.6)

or

Xd = -772 Vf + 32.144 (mm) (6.7)

171

Where

Xd = Average crack spacing at saturation with Hooked fiber

Vf = Volume fraction of fiber

The relationship obtained for the average crack width with Hooked fiber is given

by:

Wav = -0.08 Vf + 0.0043 (in) (6.8)

Wav = -2.032 Vf + 0.1092 (mm) (6.9)

where

Wav = Average crack width at saturation with Hooked fiber


or

(6.9)

Furthermore, the relation between average crack width and average crack spacing

was also found to be linear as shown in Fig. 6.24.

Figure 6.24: Average crack width versus average crack spacing for test series with

Hooked fiber

172

6.5.1.1 Calculation of Strain at Maximum Stress for Test Series with Hooked

Fiber

The strain at maximum stress can be predicted from the average crack spacing

and average crack width relation, as obtained from Eqs. (6.6 to 6.9) and the two curves

described in Fig 6.22 and Fig 6.23. The following equation is used:

d

avpc X

W=ε (6.10)

where εpc = Strain at maximum stress

Wav = Average crack width

Xd = Average crack spacing

An example for calculating the strain at maximum stress is illustrated as follows:

For Vf = 1.5%

Strain at maximum stress = d

avpc X

W=ε = 003996.0

7807.0003199.0

=

Table 6.9: Series with Hooked, Average crack spacing, Average crack width and

predicted strain

Values of strain at maximum stress are summarized in Table 6.9 It can be

observed that predicted values are comparable to the experimental values obtained from

the direct tensile test series with Hooked fiber described in Chapter 4.

173

6.5.2 Test Series with Torex Fiber

Multiple cracking in specimens with Torex fibers was extensive and the number

of cracks increased with the volume fraction of fiber as expected.

Figure 6.25: Crack formation under increasing load in test series reinforced with 1.5%

Torex steel fiber

Figure 6.25 illustrates the formation of cracks with increasing load up to

localization in a typical tensile specimen with Torex fiber.


spacing and average crack width at saturation versus the volume fraction of Torex fiber.

These were obtained in a manner similar to that described above for the Hooked fiber.

Almost linear relationships are observed. Therefore linear equations were developed.

174


saturation for test series with Torex fiber


saturation for test series with Torex fiber

The relationship obtained for the average crack spacing with Torex fiber is given by:

Xd = -65.63 Vf + 1.61 (in) (6.11)

or

Xd = -1667 Vf + 40.87 (mm) (6.12)

where

Xd = Average crack spacing in test series reinforced with Torex fiber


The relationship obtained for the average crack width with Torex fiber is given by:

Wav = -0.12 Vf + 0.0043 (in) (6.13)

where

or

Wav = -3.048 Vf + 0.1092 (mm) (6.14)

where

Wav = Average crack width in test series reinforced with Torex fiber


Furthermore, the relation between average crack spacing and average crack width

is illustrated in Fig. 6.28.

175

Figure 6.28: Average crack width versus average crack spacing at saturation in tensile

test series with Torex fiber

6.5.2.1 Calculation of Strain at Maximum Stress for Test Series with Torex

Fiber



described in Fig. 6.26 and Fig. 6.27. The following equation is used:

d

avpc X

W=ε (6.15)





For Vf = 1.5%


avpc X

W=ε = 005417.0

4966.000269.0

=

176

Table 6.10: Series with Torex, Average crack spacing, Average crack width and

predicted strain

Values of strain at maximum stress are summarized in Table 6.10. It can be


the direct tensile test series with Torex fiber described in Chapter 4.

6.5.3 Test Series with Spectra Fiber


spacing and average crack width at saturation versus the volume fraction of Spectra fiber.

These were obtained in a manner similar to that described above for the Hooked fiber.

Almost linear relationships are observed. Therefore linear equations were developed.


saturation for test series with Spectra fiber


saturation for test series with Spectra fiber

177

The relationship obtained for the average crack spacing with Spectra fiber is given by:

Xd = -30.5 Vf + 1.1 (in) (6.16)

or

Xd = -775 Vf + 27.9 (mm) (6.17)

where

Xd = Average crack spacing in test series reinforced with Spectra fiber


The relationship obtained for the average crack width with Spectra fiber is given

by:

Wav = 0.52Vf + 0.0026 (in) (6.18)

or

Wav = 12.7432 Vf + 0.06604 (mm) (6.19)

where

Wav = Average crack width in test series reinforced with Spectra fiber


Furthermore, the relation between average crack spacing and average crack width

is linear as illustrated in Fig. 6.31.

Figure 6.31: Average crack width versus average crack spacing at saturation in tensile

test series with Spectra fiber

178

6.5.3.1 Calculation of Strain at Maximum Stress for Test Series with Spectra

Fiber

Figure 6.32: Typical tensile specimens reinforced with Spectra fiber at the end of test



described in Fig. 6.29 and Fig. 6.30. The following equation is used:

d

avpc X

W=ε (6.20)





For Vf = 1.5%


avpc X

W=ε = 01597.0

6262.001.0

=

179

Table 6.11: Series with Spectra, Average crack spacing, Average crack width and

predicted strain

Values of strain at maximum stress are summarized in Table 6.11. It can be


the direct tensile test series with Torex fiber described in Chapter 4.

6.6 Modeling the Softening Response after Localization

The softening stage after localization requires special consideration. In this stage,

the elongation of a tensile specimen under load cannot be translated into a strain since it

represents the elongation of a single crack, the critical crack at localization. Therefore, a

relation between tensile stress and displacement (mainly due a single crack opening) is

applied.

Figure 6.33: Stress-displacement relation after localization

180

The post-localization stage involves the possible combination of fiber pulling out

of the matrix, fiber breaking, damage in the matrix around the critical zone, and reduction

of specimen cross-sectional area. In the experimental tests carried out in Chapters 4 and

5, the use of PVA fiber led to a very poor post-localization response. Fibers failed after a

relatively small crack opening. All the other fibers tested namely Hooked, Torex and

Spectra fibers pulled out from the critical section inducing various degrees of damage.

No fiber failure could be observed.

For a fiber pulling out from the matrix, the stress in the fiber can be computed

from the following equation assuming a circular fiber:

For a single aligned fiber:

)(4

00 uuL

d ef

−+=τ

σ (6.21)

For a randomly oriented fiber in space:

∫ ∫=

=1

2

1

0

)(cos)(),()(φ

φ

φφφφδδσeL

eef

f ddLpgLPAV

(6.22)

Figure 6.34: Localization-failure in test

series with PVA fiber

Figure 6.35: Fibers assumed to pull-out for

specimens with Hooked, Torex, and

Spectra fibers

Experimental studies on fiber reinforced cementitious composite have shown that

181

the post-peak displacement is essentially the result of the propagation-opening of the

localized crack along the notched section. The stress-displacement relationship in the

post peak region can then be evaluated as equivalent to the stress versus crack width,

assuming a single equivalent crack, with all fibers in a state of pull-out. The condition to

model is illustrated in Fig. 6.35, except that fibers are randomly oriented instead of being

aligned as shown. For specimens with a single crack having a width x, the load T at the

crack width x can be formulated as follows:

f

f

L

xL

DNLNDUT

2)2

( −==τπ

π (6.23)

where:

2

)2

( xL

L

f −= (6.24)

)

2(

)2

(

f

f

L

xL

NN

−= (6.25)

Where D = fiber diameter (assuming round fibers)

τ = average value of bond strength

T = load at the crack width x

L = average embedded length of the fibers at the crack width x

N = number of fibers having some embedded length at the crack width x

N = number of fibers located where crack width is zero

Lf = length of the fiber

From Eq. 6.23, 6.24, and 6.25 lead to the development of simplify equation,

(Kosa and Naaman, 1990). The relationship between the stress in the composite cracked

section and displacement (or crack opening) can be calculated as

2

max

)5.0

1(fL

x−=

σσ (6.26)

182

Equation (6.26) indicates that the stress becomes zero at a crack opening equal to

half the fiber length. This is theoretically correct, provided the bond remains constant

and no other deterioration occurs during pull-out. Otherwise, the stress may reach a zero

value as a crack opening smaller than half the fiber length. Kosa used a modifier K1 to

account for that effect, when his specimens were exposed to corrosion and the fibers

failed prior to complete pull-out.

Figure 6.36: Embedded length of aligned fibers bridging a crack

After extensive evaluation of Eq. (6.26) with the modifier K1using the test results

of this study, it was found that K1 was not sufficient to allow a full representation of the

stress versus displacement response; an additional parameter was needed to allow

modification of the curvature of the curve between the maximum stress and zero. After

several trials, an exponential term was added as a multiplier using a modifier K2. This

led to the development of the following equation:

xK

f

eLK

x22

1max

)5.0

1( −−=σσ (6.27)

where σ = stress at the crack width x

σmax = peak-stress

x = crack width

Lf = length of the fiber

K1 = modifier previously described with a value ranging from 0 to 1

K2 = modifier previously described.

183

With the parameters K1 and K2 Eq. (6.27) can account for several effects such as

partial fiber failure, bond decay, matrix damage, etc.

Using Eq. (6.27) to fit the average stress versus COD for the different test series

(Chapter 5) of this study leads to the best values of K1 and K2 given in Table 6.12.

Table 6.12: Value of parameters K1 and K2 for (Eq. 6.27)

Fiber

type PVA Spectra Hooked Torex

Vf K1 K2 K1 K2 K1 K2 K1 K2

0.75% 0.111 0 1 11.787 1 4.165 1 6.958

1.00% 0.085 0 1 10.354 1 2.791 1 3.413

1.50% 0.103 0 1 1.9 1 15.156 1 7.291

2.00% 0.131 0 1 1.774 1 15.839 1 5.45

Average 0.1075 0 1 6.45375 1 9.48775 1 5.778

K1Lf/2 represents the maximum displacement at which the load is equal to zero.

K2 allows to vary the slope of the normalized curves. It was analyzed from regression

analysis using an exponential function. Figure 6.36 shows the typical case for the series

with 2% Torex fibers. The three normalized curves are shown as well as the fitting curve

with K1 = 1 and K2 = 5.45; for comparison the fitting curve for K2 = 1 is also shown.

Usually, in test series reinforced with Torex, Spectra and Hooked fiber, no fiber

failure is observed and fibers pull-out to complete separation. K1 can thus be taken equal

to 1. However, for the test series with PVA fibers, fiber pull-out was not observed, and

extensive fiber failure occurred leading to a K1 value of 0.11 (Table 6.12). Then there

was no need to use K2 as the fit excellent. These two cases illustrate how flexible Eq.

(6.27) can be.

184

Figure 6.37: Example of normalized stress-displacement response of test series reinforced

with 2% Torex fiber

If the K1 and K2 values of Table 6.9 are averaged for each fiber at the four volume

fractions tested (0.75% to 2%), the following results are obtained: K2 = 5.778 for Torex

fibers, 9.488 for Hooked fiber, 6.454 for Spectra fiber, and K1 = 0.11 for PVA fiber.

These can be used as best guess values for predicting the response of other test series

with similar fibers. Figure 6.38 illustrates the predicted stress-displacement curves after

localization of the various test series for different volume fractions of fiber, and the

average curve recommended for each type of fiber. Note that the scale of the x axis for

the PVA fiber is different from that of the other fibers. Figure 6.39 compares the average

curve predicted for each fiber, using the average values of the coefficients K1 and K2 for

all fiber volume fractions between 0.75% and 2%. It can be observed that the curve with

PVA fiber indicate significant damage, while the curves for the Hooked and Spectra

fibers indicate moderate damage, and that for Torex a smaller damage. All these curves

have a curvature larger than that of the perfect parabola.

185

(a) (b)

(c)

(d)

Figure 6.38: Normalized curves after localization and their average for the test series

with: (a) PVA fiber, (b) Spectra fiber, (c) Hooked fiber, and (d) Torex fiber

186

Figure 6.39: Predicted average stress-displacement curves after localization for all test

series with notched specimens

6.7 Correlation between Direct Tensile Tests and Stress versus Crack

Opening Displacement Tests

Figure 6.40: Correlation between strain and maximum stress from a direct tensile test and

displacement at maximum stress from a notched prism test

The purpose of this section is to show that it is possible to correlate the strain at

maximum stress from a direct tensile test and the displacement a maximum stress from a

notched tensile prism test. This is achieved by estimating the equivalent number of

187

cracks that makes the composite look like a chain with a number of links (Fig. 6.40). An

energy approach will be used to estimate the number of cracks. The assumptions for

estimating the number of cracks, thus the average crack spacing are as follows.

Figure 6.41: Zone of cracking influence around

main crack

1. Any crack can become a

localized crack.

2. The energy consumed by the

crack, in the crack-opening

displacement experiment, is

the same as that consumed by

a crack in the tensile prism.

Figure 6.42: Correlation between “zone of cracking” from a stress-crack opening

displacement test and a direct tensile test (Dogbone test)

Let’s define Ene as the energy required to create a group of cracks in the notched

specimen around the influence zone up to crack saturation. This energy is assumed to

creat a group of craks as a cluster in a HPFRCC strain hardening material.

188

The surface energy to create a length of crack (for same width in the specimen) is

equal to

ncc

nes xnA

EE = (6.28)

Where Es = surface energy (psi/in , MPa/mm)

Ene = energy at the end of cracking saturation in a stress versus crack-

opening displacement specimen (Notched specimen)

Ac = cross-sectional area of stress crack-opening displacement specimen

(Notch specimen, equal to 2 in2, 1290 mm2 in this study)

nnc = number of equivalent cracks (representing the cluster of cracks) in

a notched specimen

Let us consider a direct tensile test specimen (Dogbone). Each group of clustered

cracks that occur during the multiple cracking strage is assumed to consume as much

energy as that consumed in the notched tensile prism, that is, Ene. Multiple cracking will

continue in the direct tensile test specimen until the total energy nsatEne, where nsat is the

number of cracks at saturation, becomes equal to the energy consumed at the end of

multiple cracking, Ede. (See also Fig. 3.14).

This energy equation is expressed as follows:

denesat EEn ≤ (6.29)

Where nsat is the number of assumed crack clusters in a direct tensile test

specimen.

Figure 6.43: Schematic stress-strain curve and crack clusters

189

Localization will occur when the energy required to open the critical crack is

smaller than the total energy required to create a new crack. So nsatEne cannot be higher

than Ede, thus

ne

desat E

En = (6.30)

If we assume that G is the gauge length of a given tensile prism (equal to 7 inches,

178 mm as used in the direct tensile tests described in Chapter 4), nnc = the number of

equivalent cracks in a notched specimen, then number of cracks in direct tensile specimen

(Dogbone specimen) at the saturated state, can be calculated by

Number of cracks ne

denc

EGEn

= (6.31)

(a) (b)

Figure 6.44: Definition of Ede ,Ene, Dde, and Dne

Results from direct tensile tests (Dogbone) and from the notched prism tests were

analyzed and correlated. A relation between the number of cracks, energy at the end of

the multiple cracking stage from the Direct tensile test (Ede), and energy at the end of the

cluster cracking stage of the Notched tensile test (Ene) was established.

Additionally, as a crack has two opposing surfaces, the stress versus crack

190

opening energy can be considered equal to twice the surface energy of out of the

composite, γc. Thus:

Ene= 2γc (6.32)

Where γc is the surface energy of the composite.

The surface energy of the composite γc can be taken equal to the sum of the pull-

out energy γp consumed by the fibers during pull-out, and the surface energy of the

matrix γm, that is:

γc = γp+ γm (6.33)

As the surface energy due to fiber pull-out is substantially larger than the surface

energy of the matrix, γm can be neglected and the value of γp can be used as a first

approximation to estimate the surface or fracture energy of the composite. Note that this

approach is justified since in all notched test series, except when PVA fibers were used,

general fiber pull-out was observed.


Figure 6.45: Typical failure crack in a

tensile specimen reinforced with PVA

fiber

Figure 6.46: Cracking in a notched prism

with PVA fiber

191

If we take the energy absorption of the Direct tensile test series (Ede) and the

energy absorption of the Notched prism series (Ene) up to the maximum tensile stress

(area under the stress-strain orstress-displacement curve), and compare them together,

they are almost equal (Ede = Ene, Ede / Ene = 1). Figure 6.47 summarizes the results. They

suggest that only one single crack could be occurred. This is indeed what was observed as

shown in Figs. 6.45 and 6.46.

Vf PVA

(Ede/Ene)

0.75 0.8948

1 1.000735

1.5 0.999193

2 0.99551

PVA (Ede/Ene)

0.89481.0007 0.9992 0.9955

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.75 1 1.5 2Fiber volume fraction (%)

Ede /E

ne

Figure 6.47: Ede / Ene for test series with PVA fiber

192



tensile specimen reinforced with Spectra

fiber test


with Spectra fiber

Figures 6.48 and 6.49 show typical cracking in a direct tensile test specimen and

in a notched tensile prism, respectively. The length of total cracking in a notched prism

is calculated from photographic records and divided by the specimen’s width (2 in or

50.8 mm) to arrive at an equivalent number of cracks. This is called here a crack cluster

for the notched prism. Typical results, that is the equivalent number of cracks for a crack

cluster in a notched prism, are plotted in Fig. 6.50 for the different fiber volume fractions

tested. It can be observed, that the number of crack clusters ranged from 2.6 to 6.7

cracks, essentially increasing with the volume fraction of fiber up to 1.5% then tapering

off. Also, unlike notched specimens with a plain matrix where only one crack equivalent

is observed, the number of equivalent cracks here is significant. Note that in comparison,

only one crack equivalent was observed with PVA fiber.

The energy absorption of the Direct tensile test series (Ede) and the energy

absorption of the Notched prism series (Ene) up to the maximum tensile stress (area under

193

the stress-strain or stress-displacement curve), are calculated for the different volume

fractions of fibers. Figure 6.51 illustrates the variation of the ratio Ede/Ene for the different

volume fractions of fiber used. It can be observed that the ratio does not show any

particular trend and could be considered to remain about the same.

Vf

Number

of

Cracks

0.75 2.7 1 3.2

1.5 6.8 2 6.2

Number of Cracks in Notch HPFRCC reinforced Spectra

2.73.2

6.86.2

0

1

2

3

4

5

6

7

8


Num

ber o

f Cra

cks

Figure 6.50: Number of equivalent cracks in a cluster in notched test series with

Spectra fiber

Vf Spectra

(Ede/Ene)

0.75 2.916

1 3.449

1.5 2.711

2 2.860

Series of Spectra (Ede/Ene)

2.9163.449

2.711 2.860

00.5

11.5

22.5

33.5

44.5

5


Ede /E

ne

Figure 6.51: Ede / Ene for test series with Spectra fiber

194

By multiplying the number of equivalent cracks found in the Notch specimen for

a crack cluster by the energy ratio Ede / Ene, the number of cracks in the Direct tensile test

specimen can be determined. Predicted values are shown in Fig. 6.52 for the different

volume fractions of fiber tested and compared to experimental observations from

photographic records. As seen in Fig. 6.52, good agreement is observed.

Vf

Number

of Cracks

Predicted

0.75 7.8 1 10.9 1.5 18.3 2 17.8

HPFRCC Reinforced SpectraNumber of Cracks in Direct Tensile Specimen

7.1

10.9 11.213.3

7.8

11.0

18.3 17.8

0

5

10

15

20

0.75 1 1.5 2Volume Fraction of Fiber (%)

Experiment

Predict

Figure 6.52: Comparison of the number of cracks in direct tensile test series with

Spectra fiber (Dogbone test) obtained experimentally and estimated from the energy

method






for the notched prism. Typical results, that is the equivalent number of cracks for a crack

cluster in a notched prism, are plotted in Fig. 6.55 for the different fiber volume fractions

tested. It can be observed, that the number of crack clusters ranged from 2.7 to 6.2

cracks, essentially increasing with the volume fraction of fiber up to 1% then tapering off.

Here also, unlike notched specimens with a plain matrix where only one crack equivalent

is observed, the number of equivalent cracks is significant. Note that in comparison, only

one crack equivalent was observed with PVA fiber.

195


tensile specimen reinforced with Hooked

fiber


with Hooked fiber

196

Vf

Number

of

Cracks

0.75 2.7 1 5.2

1.5 5.0 2 4.2

Number of Cracks in Notch HPFRCC reinforced Hooked

2.7

5.2 5.04.2

0

1

2

3

4

5

6

7


Num

ber o

f Cra

cks

Figure 6.55: Number of equivalent cracks in a cluster in notched test series with

Hooked fiber





volume fractions of fiber used. It can be observed that the ratio can be considered to

increase with the volume fraction of fiber.

197

Series of Hooked (Ede/Ene)

0.615

1.7231.472

2.189

0

0.5

1

1.5

2

2.5

3


Ede /E

neFigure 6.56: Ede / Ene for test series with Hooked fiber




volume frations of fiber tested and compared to experimental observations from

photographic records. As seen in Fig. 6.57, very good agreement is observed.

Vf Hooked

0.75 1.7

1 8.9

1.5 7.3

2 9.2

HPFRCC Reinforced HookedNumber of Cracks in Direct Tensile Specimen

2.0

7.2

9.0

10.4

1.7

8.9

7.3

9.2

0

2

4

6

8

10

12


Experiment

Predict

Figure 6.57: Comparison of the number of cracks in direct tensile test series with

Hooked fiber (Dogbone test) obtained experimentally and estimated from the energy

method

198






for the notched prism. Typical results, that is, the equivalent number of cracks for a

crack cluster in a notched prism, are plotted in Fig. 6.60 for the different fiber volume

fractions tested. It can be observed, that the number of crack clusters was consistently

high at all values of Vf with an average of 5 cracks.


tensile specimen reinforced with Torex fiber


with Torex fiber

199

Vf

Number

of

Cracks

0.75 5.9 1 5.0

1.5 5.8 2 4.1

Number of Cracks in Notch HPFRCC reinforced Torex

5.9

5.05.8

4.1

0

1

2

3

4

5

6

7


Num

ber o

f Cra

cks

Figure 6.60: Number of equivalent cracks in a cluster in notched test series with Torex

fiber





volume fractions of fiber used. It can be observed that the ratio can be considered to

increase with the volume fraction of fiber.

Series of Torex (Ede/Ene)

1.099 0.864

3.908

4.833

0

1

2

3

4

5

6


Ede /E

ne

Figure 6.61: Ede / Ene for test series with Torex fiber

200




volume fractions of fiber tested and compared to experimental observations from

photographic records. Reasonably good agreement is observed.

Vf Torex

0.75 6.5 1 4.3

1.5 22.7 2 19.8

HPFRCC Reinforced TorexNumber of Cracks in Direct Tensile Specimen

6.3 6.9

14.1

19.4

6.54.3

22.719.8

0

5

10

15

20

25


Experiment

Predict

Figure 6.62: Comparison of the number of cracks in direct tensile test series with Torex

fiber (Dogbone test) obtained experimentally and estimated from the energy method

Table 6.13: Surface energy and area under the curve obtained from tests (US-units)

201

Table 6.14: Surface energy and area under the curve obtained from tests (SI-units)

Table 6.15: Energy ratio and influence unit /length of crack-opening displacement test

6.8 Outline of Stress-Strain Computations

The proposed stress-strain and stress-displacement computational method is

presented in four steps (Fig. 6.63). The first step is to find the main three unknown

parameters (ultimate strain, first cracking stress and ultimate stress). The second step is to

calculate the crack spacing and crack spacing at saturation. The third step is to obtain the

crack opening from COD (Notched prism test).

Finally, the forth step illustrates by example a method to calculate the strain at

post cracking (εp) and the strain at maximum stress (εpc) as well as the displacement at the

localization stage based on the linking assumption.

An example of verification of the proposed model using a composite reinforced

with 1.5% Torex 1.5% is presented in Section 6.9.

202

Figure 6.63: Flowchart of proposed post-cracking model

203

6.9 Verification of the Model 6.9.1 Tensile Specimen with Torex Fiber

The purpose of this section is to apply the tensile stress-strain model using

information obtained from this research. For this a typical composite reinforced with

1.5% Torex fiber is considered.

Material properties: σmu = 1.252 MPa, τ = 6.84 MPa, L/d = 100 , Vf = 1.5%, Ec =

13890 MPa

From Fig. 6.10: α1 α2 = 0.0937

Thus:

σcc = σmu (1-Vf) + α1 α2 τ Vf L / d

= 1.252 x (1-0.015) + 0.0937 x 6.84 x 0.015 x 100

= 1.233 + 0.961

= 2.194 MPa

εcc = c

cc

Eσ

= 410579.113890

194.2 −= x

From Eq. (6.6): 971.0601.26 +−= fpc Vλ

λpc = -26.601 x 0.015 + 0.971

= 0.572

σpc = λpc τ Vf L/d

= 0.5724 x 6.84 x 0.015 x 100

= 5.872 MPa

204

Find εpc: Xd = -1667 Vf + 40.873 (6.12)

= -1667 x 0.015 +40.873

= 15.86 mm

Wav = -3.05 Vf + 0.1092 (6.14)

= -3.05 x 0.015 +0.1092

= 0.0635 mm

d

avpc X

W=ε (6.10)

869.15

0635.0=

= 0.004 Response curve after localization

xK

f

eLK

x22

1max

)5.0

1( −−=σσ (6.27)

From Table 6.12, K1 = 1, K2 = 5.778 (Average)

Gauge length = 7 inches, 177.8 mm, Fiber length = 30 mm, σmax = 5.873 MPa Table 6.16: Computations of stresses and displacements for the example specimen

205

Figure 6.64: Comparison of predicted versus experimental stress-strain curve

6.9.2 Other Model Predictions

The above developed model was systematically applied to the different tensile test

series experimentally tested in this research. The predicted tensile stress-strain response

was simplified to getting the first cracking point, the maximum post-cracking or ultimate

point, and the softening branch after localization. The numerical results obtained are

summarized in Table 6.17. The results are plotted in Figs. 6.65 to 6.71 and are self

explanatory. They could be compared to the experimental curves described in Chapters 4

and 5. Note that a range of values is shown for the ultimate point (maximum stress point)

to illustrate the influence of the volume fraction of fiber from 0.75% to 2%.

206

Figure 6.65: Schematic stress-strain response of FRC reinforced with PVA fiber (Direct tensile test)

Figure 6.66: Schematic stress-displacement response of FRC reinforced with PVA fiber

(Notch tensile test)

207

Figure 6.67: Schematic stress-strain response of FRC reinforced with Spectra fiber

(Direct tensile test)

Figure 6.68: Schematic stress-strain response of FRC reinforced with Hooked fiber


208

Figure 6.69: Schematic stress-displacement response of FRC reinforced with Hooked

fiber (Notch tensile test)

Figure 6.70: Schematic stress-strain response of FRC reinforced with Torex fiber


209

Figure 6.71: Schematic stress-displacement response of FRC reinforced with Torex fiber (Notch tensile test)

Table 6.17: Summary of computations for the stress-strain curves of HPFRCC tensile specimens (a, b, c, d, e, f) First Cracking Stress σcc = σmu (1-Vf) + α1 α2 τ Vf L / d (6.1) (a) Fiber First Cracking Stress PVA α1 α2 = 0.5204 :Ranged 0.75%-2.0%

Spectra α1 α2 = 0.0.0424 :Ranged 1.0% - 2.0%

Hooked α1 α2 = 0.2947 :Ranged 1.0% - 2.0%

Torex 0937.021 =αα :Ranged 1.0%% - 2.0%

210

Maximum Stress σpc = λpc τ Vf L/d (6.2) (b) Fiber Maximum Stress PVA 0933.27074.0 +−= fpc Vλ :Ranged 0.75%-2.0% (6.34)

Spectra 6271.0799.20 +−= fpc Vλ :Ranged 0.75%-2.0% (6.3)

Hooked 5836.2554.55 +−= fpc Vλ :Ranged 1.0%-2.0% (6.4)

Torex 9714.0601.26 +−= fpc Vλ :Ranged 0.75%-2.0% (6.5)

Crack spacing (US-units, in) (c) Fiber Average Crack Spacing Spectra Xd = -30.504 Vf + 1.0972 :Ranged 0.75% - 2.0% (6.16)

Hooked Xd = -30.4 Vf + 1.2655 :Ranged 1.0% - 2.0% (6.6)

Torex Xd = -65.63 Vf + 1.6092 :Ranged 0.75% - 2.0% (6.11)

211

Crack spacing (SI-units, mm) (d) Fiber Average Crack Spacing Spectra Xd = -774.802 Vf + 27.8689 :Ranged 0.75% - 2.0% (6.17)

Hooked Xd = -772.2 Vf + 32.1437 :Ranged 1.0% - 2.0% (6.7)

Torex Xd = -1667 Vf + 40.8737 :Ranged 0.75% - 2.0% (6.12)

Crack Width (US-units, in) (e) Fiber Average Crack Width Spectra Wav = 0.5017 Vf + 0.0026 :Ranged 0.75%-2.0% (6.18)

Hooked Wav = -0.08 Vf + 0.0043 :Ranged 1.0%-2.0% (6.8)

Torex Wav = -0.12 Vf + 0.0043 :Ranged 0.75%-2.0% (6.13)

Crack Width (SI-units, mm) (f) Fiber Average Crack Width Spectra Wav = 12.7432 Vf + 0.06604 :Ranged 0.75%-2.0% (6.19)

Hooked Wav = -2.032 Vf + 0.1092 :Ranged 1.0%-2.0% (6.9)

Torex Wav = -3.05 Vf + 0.1092 :Ranged 0.75%-2.0% (6.14)

212


A model based on a combination of composite mechanics and experimental

observations to predict the stress-strain response of fiber reinforced cement composites in

the post-cracking state, and particularly the multiple cracking stage was developed. The

influence of various parameters was investigated. Good agreement was found between

the model predictions for stress-strain relation and experimental results. The following

conclusions are drawn.

1. Predictive equation derived from mechanics of composites was used for the stress

at first cracking and the maximum post-cracking stress. From the experimental

results obtained, prediction equations were suggested for the coefficients in these

equations

2. Predictions equations for average crack spacing and average crack width for

HPFRCC tensile specimens reinforced with Spectra, Hooked, and Torex fibers

were derived based on experimental observations.

3. A method to predict the strain at maximum stress, from crack width and crack

spacing is suggested. A analytical equation was developed to predict the response

of the composite in tension after localization. The equation depends on two

parameters only, and allows to simulate damage as well as variable curvature in

the shape of the curve. The model produced good approximation of the

localization behavior obtained experimentally.

4. A correlation between the direct tensile test (Dogbone test) and stress-crack

opening displacement test (Notch test) was established. The proposed method was

found to have a good agreement between prediction and experiment.

213

Initial stage Stress = 0 psi Displacement = 0 in Initiation of first crack Stress = 295.97 psi Displacement = 0.000533 in Expanding of first crack Stress = 661.038 psi Displacement = 0.002135 in Maximum stress Stress = 781.7 psi Displacement = 0.00979 in Localization Stress = 688.265 psi Displacmen = 0.0196 in

Figure 6.72: Crack formation under increasing load in test series reinforced with 0.75% Torex steel fiber (Notch test)

214

CHAPTER 7

TENSILE VARIABILITY OF FIBER REINFORCED CEMENTITOUS MATERIALS

7.1 Introduction

The design of structures composed of fiber reinforced material should be based on

the statistical consideration of FRCC properties. This chapter is therefore dedicated to a

discussion of the observed statistical variability in HPFRCC response. The parameters

considered include toughness, tensile strength (σpc), energy, ductility, and strain at

maximum stress (εpc).

Generally, the variability of HPFRCC response is strongly influenced by the

fabrication process, sampling, and testing. Geometry of the fiber in relation to the mould

size causes preferred orientation of the fibers near the surface of the mould, which is

different from that in the body of the composite. The orientations within the mass of the

composite depend on many factors such as the method of placing, flow characteristics of

the mortar and fiber, and the type and degree of compaction. There are other factors that

tend to increase the statistical variations in the properties of fiber-reinforced cement

composites. These factors include

1. variations in the concentration of fibers at different locations inside the matrix;

2. variability in the interfacial bond properties;

3. relatively low workability (compatibility) of fibrous composites in some mixes,

which may leave a system of entrapped air with different local concentrations

inside the mix.

215

Swamy and Stavirdes reported the results of a study on statistical variations in the

flexural toughness of steel fiber reinforced concrete. This study was based on limited test

results and indicated relatively large variations in toughness. They noticed that flexural

toughness could not be correlated to flexural strength. In their test results, the relatively

large variations in flexural toughness were attributed to high variability in the fiber

debonding process in steel fiber reinforced concrete (which determines toughness

characteristics).

In the test results reported in this chapter, the coefficient of variation for steel

fiber concrete within a batch and between batches was well below the 5% confidence

level and of the same order as those obtained for plain concrete. It was therefore possible

to conclude that in steel fiber reinforced concrete, where good quality control is exercised

throughout fabrication, the number of test specimens for obtaining a reliable average

need to be no more than that required for plain concrete. Additionally, it was indicated by

Swamy and Stavirdes that the slight increase observed in compressive strength in the

presence of steel fibers is insignificant in light of the random variations in test results.

Statistical Parameters

The normal distribution, also known as the Gaussian distribution, has a

probability density function given by

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−=2

21exp

21)(

σμ

πσxxf x ∞<<∞− x (7.1)

where μ and σ are the parameters of the distribution, which are also the mean and

standard deviation, respectively, of the distribution. Figure 7.1 and 7.2 show the effects of

μ and σ on the probability density function and cumulative distribution function.

216

Figure 7.1: Probability density function

with different μ and σ Figure 7.2: Cumulative distribution

function

The coefficient of variation (COV) is a normalized measure of dispersion of a

probability distribution. It is defined as the ratio of the standard deviation to the mean.

σμ

=COV (7.2)

The confidence interval is an interval estimate of a sample parameter. Instead of

estimating the parameter by a single value, an interval likely includes the parameter as

stated. How likely the interval is to contain the parameter is determined by the confidence

level. Increasing the confidence level will widen the confidence interval. In contrast,

decreasing the confidence level will narrow the confidence interval.

Figure 7.3: 95% Confidence interval in normal distribution

The goodness of fit of a statistical model describes how well it fits a set of

observations. Measures of goodness-of-fit typically summarize the discrepancy between

observed values and the values expected under the model in question. Such measures can

be used in statistical hypothesis testing, e.g. to test for normality of residuals. In this

work, three fitting test were used, Cramer-von Mises test, Watson, and Anderson-Darling

test.

217

Six series were considered in this study, including specimens reinforced Spectra

2.0%, 1.5%, specimen reinforced Hooked 2.0%, 1.5%, and specimen reinforced Torex

2.0%, 1.5%. The number of specimens per series ranged from 13 to 28 specimens. A total

of 120 samples were employed in this analysis. The significant parameters considered are

maximum stress, strain at maximum stress, strain at the end of multiple cracking stage,

energy at maximum stress, and energy at the end of multiple cracking stage. A summary

of the statistical analysis conducted is displayed in Table 7.1. The toughness parameter

(ductility) is defined as the area under the curves, up to localization as displayed in Fig.

7.4. Since, in the case of HPFRCC reinforced Spectra, the influence of the multiple

cracking area is large and dominates the localization behave ior, the energy at the end of

multiple cracking stage was used as a representation for toughness.

Figure 7.4: Energy at localization start: area under the curve, toughness

Figure 7.5: Multiple cracking stage in notch HPFRCC reinforced Spectra fiber

218

Table 7.1: Diagram for analysis of tensile histogram and normal probability curve

7.2 Confidence Interval (CI)

The CI can be used to describe how reliable HPFRCC tensile results are. For a set

of test results, the outcome of the maximum tensile stress might be 95% of maximum

stress in a certain direct tensile test in each series. A 95% confidence interval implies that

the results have a probability of occurrence ranging 2.5% to 97.5%. All other things

being equal, a survey result with a small CI is more reliable than a result with a large CI.

(a) (b)

Figure 7.6: (a) Direct tensile curve of HPFRCC reinforced Torex 1.5% and (b) its

corresponding confidence interval

219

(a) (b)

Figure 7.7: (a) Direct tensile curve of HPFRCC reinforced Torex 2.0% and (b) its


(a) (b)

Figure 7.8: (a) Direct tensile curve of HPFRCC reinforced Hooked 1.5% and (b) its


220

(a) (b)

Figure 7.9: (a) Direct tensile curve of HPFRCC reinforced Hooked 2.0% and (b) its


The confidence interval curves for 95 percent are displayed in Fig. 7.6 to Fig. 7.9

for HPFRCC reinforced Hooked and Torex, at Vf of 1.5% and 2.0%. More variation

represents a large area of variability in any testing range. The widest range of confidence

interval is for maximum stress, for 2.0% specimen reinforced Hooked. In this case, the

upper bound is 1276 psi (8.798 MPa), while the lower bound is 372 psi (2.565 MPa). The

difference between both bounds is 343 percent of the lower bound.

7.3 Coefficient of Variation (COV) Analysis

This section discusses the COV in the statistical data. Specimen reinforced with

PVA and Spectra at volume fraction of 0.75% were not included in this analysis since the

number of specimen is insufficient.

221

Figure 7.10: Coefficient of variation, maximum stress

Figure 7.10 presents the coefficient of variation for maximum stress. From the

graph, the coefficient of variation increases with increase in volume fraction of fiber,

except in the case of HPFRCC Reinforced Torex, where COV appear to be about

constant. Overall, the coefficient of variation ranged from 11.32% in HPFRCC reinforced

Hooked 0.75% to 32.48% in HPFRCC reinforced Hooked 2.0%. The average COV of

HPFRCC reinforced Spectra, is 20.48%, COV of HPFRCC reinforced Hooked is 19.62%,

and HPFRCC reinforced Torex is 18.51%

Figure 7.11: Coefficient of variation, energy at maximum stress

222

Figure 7.11 presents the coefficient of variation of toughness (energy at maximum

stress). The value ranges from 33.86% in HPFRCC reinforced Torex 2.0% to 78.96% in

HPFRCC reinforced Hooked 1.0%. In HPFRCC reinforced Hooked, when volume

fraction of fiber increase, the coefficient of variation increase. However, in HPFRCC

reinforced Torex, increasing the volume fraction of fiber leads to decrease in coefficient

of variation. Spectra fiber does not see much change in terms of variability, is in the

range of 60 percent. The average COV of HPFRCC reinforced Spectra, is 58%, COV of

HPFRCC reinforced Hooked is 63%, and HPFRCC reinforced Torex is 52%

Figure 7.12: Coefficient of variation, strain at maximum stress

The observed coefficient of variation for the strain at maximum stress is presented

in Fig 7.12. From this figure, it can be seen that the coefficient of variation ranged from

28% for HPFRCC reinforced Torex 2.0% to 85% in HPFRCC reinforced Hooked 1.0%.

It is clear that the coefficient of variation of strain at maximum stress is not dependent on

the volume fraction of fiber. The average COV of HPFRCC reinforced Spectra is 45%,

The average COV of HPFRCC reinforced Hooked is 60%, and also the average

coefficient of Torex is 38%.

223

Coefficient of Variation (COV)

20.5

58.8

44.5

19.6

63.060.1

18.5

52.3

37.6

52.08

68.41

00

10

20

30

40

50

60

70

80

1 2 3

Spectra

HookedTorex

Mortar (No fiber)

COV (%)

Maximum Stress Strain at Maximum stress Toughness

Figure 7.13: Comparison of coefficient of variation with different fiber types

Figure 7.13, compares the COV for the maximum stress, strain at maximum

stress, and toughness for all series. Clearly the variation in maximum stress is least

(average less than 20%) of the 3 parameter considered. It is also evident that adding fiber

increased the variability in maximum stress of the mortar, which is higher than that for

the HPFRCC specimens (52% versus and average of 20% for unreinforced specimens).

The strain of maximum stress has a high COV, while the COV for the toughness is

moderate. In addition to this, comparing by type of fiber (Spectra, Hooked and Torex),

the HPFRCC reinforced Torex, the coefficient of variation is lowest, while HPFRCC

reinforced Spectra and Hooked, the results are higher.

7.4 Goodness-of-Fit

Table 7.2 illustrates the goodness-to-fit test of normal probability of the data

considered in this thesis. In the results, a majority of the investigated parameters can pass

95% level of confidence for the normality test. Table 7.3 and 7.4 present a list of the

average tensile test results for HPFRCC specimens. In some cases, the number of

224

specimens is too low to be investigated. In order to analyze statistical parameters, at least

13 specimens are required, therefore the specimen in series of Spectra 1.5% and 2.0%,

Hooked 1.5% and 2.0%, Torex 1.5% and 2.0% were examined.

Table 7.2: Normality test results at 95% confidence level

Maximum stress, strain at maximum stress, energy at maximum stress (Torex and

Hooked), and energy at the end of multiple cracking stage (Spectra) are mechanical

properties of HPFRCC that were statistically analyzed in this investigation. The

distribution of all the sets of results is represented graphically by using a histogram.

Figure 7.16 to 7.33 shows the distribution of the maximum tensile stress, strain at the end

of multiple cracking stage, strain at maximum stress, energy at the end of multiple

cracking stage (Spectra), and energy at maximum stress (Torex and Hooked) of tensile

experiments. The histograms give an idea of the scatter of the results; whereas the

cumulative diagrams give the probability of results being below any given value.

225

The normal probability curves (Gaussian distribution curves) superimposed on all

the histograms were obtained by using the means and standard deviations for all the sets

of results. The height of the frequency distribution curve was computed by Statistica

software, adjusted to fit the conditions of each set of results. The normal probability

curves have also been drawn on normal probability paper as shown in this chapter in

which they appear as straight lines. In some series, departures from normal distribution

were detected. Following is a detailed discussion of the statistical data for the parameters

of interest.

Table 7.3: Statistical parameters of HPFRCC specimens (US-units)

226

Table 7.4: Statistical parameters of HPFRCC specimens (SI-units)

227

Maximum Tensile Strength – HPFRCC Reinforced Spectra 1.5%

(a) (b)

(c)

Figure 7.14: (a) Distribution of maximum stress, (b) cumulative histogram, and (c)

normal probability plot for HPFRCC reinforced Spectra 1.5%

The sample mean of these measurements is 402 psi (2.77 MPa), the standard

deviation is 89.18 psi (0.615 MPa), and the coefficient of variation is 22 percent. The 95

percent Confidence Interval regarding the mean is 227 psi (1.57 MPa) to 576 psi (3.97

MPa). The goodness-to-fit tests confirmed the normality of the sample distribution for

tensile strength test results at a 95 percent Confidence Interval. The distribution of the

results is shown in Fig. 7.14 and the normal curve overlapping the histogram presents an

228

indication of the normality of sample distribution. Figure 7.14-c shows the normal

probability plot of the tensile strength test results. This figure indicates that about 70

percent of the results are above 450 psi (3.103 MPa), and above 90 percent are above 550

psi (3.792 MPa). The cumulative distribution of the measurements is close to a straight

line, which further supports the conclusion that maximum tensile strength test results are

normally distributed.

(a) (b)

(c)

Figure 7.15: (a) Distribution of energy at the end of multiple cracking stage,

(b) cumulative histogram, and (c) normal probability plot for HPFRCC reinforced

Spectra 1.5%

229

Tensile Toughness – HPFRCC Reinforced Spectra 1.5%

The sample means of tensile toughness (energy at the end of multiple cracking

stage), was 5.96 psi, (0.041 MPa) and the corresponding standard deviations were 4.05

psi (0.028 MPa). The coefficient of variation was 68.03 percent and the 95 percent

Confidence Interval was certainly not greater than 13.9 psi (0.096 MPa). The normal

probability plot shown in figure 7.15-c shows that the test data does not fit a straight line,

indicating a deviation from normal distribution.

Strain at the End of Multiple Cracking Stage – HPFRCC Reinforced Spectra 1.5%

(a)

(b)

(c)

Figure 7.16: (a) Distribution of strain at the end of multiple cracking stage,


Spectra 1.5%

230

Figure 7.16 presents the tensile strain at the end of the multiple cracking stage of

HPRCC reinforced Spectra specimens; at this point, the starting of the localization phase

begins. The mean was 0.0162, the standard deviation was 0.007875, and the

corresponding coefficient of variation was 48.57 percent. The 95 percent confidence

interval was 0.000765 to 0.0316.

Maximum Stress – HPFRCC Reinforced Spectra 2.0%

(a) (b)

(c)

Figure 7.17: (a) Distribution of maximum stress, (b) cumulative histogram,

and (c) normal probability plot for HPFRCC reinforced Spectra 2.0%

231

The sample mean of these measurements is 466.13 psi (3.214 MPa), the standard

deviation is 123.93 psi (0.854 MPa), and the coefficient of variation is 26.59 percent. The

95 percent confidence interval with respect to the mean is 223 psi (1.538 MPa) to 709 psi

(4.888 MPa).

The normal probability plot shown in figure 7.17-c, shows that the test data do not

fit a straight line, indicating a deviation from normal distribution. The goodness-to fit

tests confirmed at a 95 percent confidence interval, the distribution is not normal.

Tensile Toughness – HPFRCC Reinforced Spectra 2.0%

(a)

(b)

(c)

Figure 7.18: (a) Distribution of energy at the end of multiple cracking stage,


Spectra 2.0%

232

The sample means was 5.97 psi (0.041 MPa), and the corresponding standard

deviation was 3.37 psi (0.023 MPa). The coefficient of variation was 56.51 percent, while

the 95 percent confidence interval was not more than 12.58 psi (0.087 MPa). The normal

probability plot shown in figure 7.18-c, shows that the test data does not fit a straight line,


Strain at Maximum Stress – HPFRCC Reinforced Spectra 2.0%

(a) (b)

(c)

Figure 7.19: (a) Distribution of strain at the end of multiple cracking stage,


Spectra 2.0%

233

Figure 7.19 presents the tensile strain at the end of the multiple cracking stage of

HPRCC reinforced Spectra specimens, starting at the localization phase. The mean was

0.0143 and the corresponding coefficient of variation was 40.35 percent. The standard

variation was 0.004171, while the 95 percent confidence interval was 0.006125 to

0.02248

Maximum Tensile Strength – HPFRCC Reinforced Hooked 1.5%

(a) (b)

(c)


and (c) normal probability plot for HPFRCC reinforced Hooked 1.5%



234

95 percent Confidence Interval regarding the mean is 374.61 psi (2.583 MPa) to 839.87

psi (5.791 MPa).

The normal probability plot shown in figure 7.20 shows that the test data does not



Tensile Toughness – HPFRCC Reinforced Hooked 1.5%

(a)

(b)

(c)

Figure 7.21: (a) Distribution of energy at maximum stress, (b) cumulative histogram,

And (c) normal probability plot for HPFRCC reinforced Hooked 1.5%

235

The sample mean was 2.0 psi (0.014 MPa), and the corresponding standard

deviation was 1.17 psi (0.008 MPa). The coefficient of variation was 58.81 percent and

the 95 percent confidence interval was no more than 4.29 psi (0.03 MPa).

Strain at Maximum Stress – HPFRCC Reinforced Hooked 1.5%

(a) (b)

(c)

Figure 7.22: (a) Distribution of strain at maximum stress, (b) cumulative histogram, and

(c) normal probability plot for HPFRCC reinforced Hooked 1.5%

Figure 7.22 presents the tensile strain at maximum stress of HPRCC reinforced

Hooked specimens, starting at the localization phase. The mean was 0.004232 and the

corresponding coefficient of variation was 54.03 percent. The standard deviation was

0.002287, while the 95 percent confidence interval was certainly no more than 0.008715.

236

Maximum Tensile Strength – HPFRCC Reinforced Hooked 2.0%

(a) (b)

(c)


and (c) normal probability plot for HPFRCC reinforced Hooked 2.0%

The sample mean of these measurements is 838 psi (5.778 MPa), the standard


95 percent confidence interval about the mean is 304.51 (2.1 MPa) to 1371.5 psi (9.456

MPa).

The goodness-to-fit tests confirmed the normality of the sample distribution for

tensile strength test results at 95 percent confidence interval. The distribution of the


indication of the normality of sample distribution.

237

Tensile Toughness – HPFRCC Reinforced Hooked 2.0%

(a) (b)

(c)

Figure 7.24: (a) Distribution of energy at maximum stress, (b) cumulative histogram, and


The sample means were 2.77 psi (0.019 MPa), and the corresponding standard

deviations were 1.88 psi (0.013 MPa). The coefficient of variation was 68.14 percent.

The 95 percent confidence interval was not more than 6.45 psi (0.044 MPa). The normal

probability plot shown in Fig. 7.24 shows that the test data does not fit a straight line,


238

Strain at Maximum Stress – HPFRCC Reinforced Hooked 2.0%

(a) (b)

(c)




Hooked specimen, starting of the localization phase. The means were 0.00416, the

corresponding coefficient of variation were 55.37 percent. The standard deviation was

0.002303. The 95 percent confidence interval was no more than 0.008674

239

Maximum Tensile Strength – HPFRCC Reinforced Torex 1.5%

(a) (b)

(c)


and (c) normal probability plot for HPFRCC reinforced Torex 1.5%



95 percent confidence interval about the mean is 404 psi (2.785 MPa) to 1093 psi (7.536

MPa).

The goodness-to-fit tests confirmed the normality of the sample distribution for

tensile strength test results at 95 percent confidence interval. The distribution of the

240


indication of the normality of sample distribution.

Tensile Toughness – HPFRCC Reinforced Torex 1.5%

(a)

(b)

(c)


(c) normal probability plot for HPFRCC reinforced Torex 1.5%


deviations were 1.52 psi (0.01 MPa). The coefficient of variation was 51.03 percent. The

95 percent confidence interval was 0.0008 psi to 5.96 psi (0.041 MPa). The normal

probability plot shown in Fig. 7.27 shows that the test data does not fit a straight line,


241

Strain at Maximum Stress – HPFRCC Reinforced Torex 1.5%

(a) (b)

(c)




Torex specimen, starting of the localization phase. The means were 0.003773, the

corresponding coefficient of variation was 45.2 percent. The standard deviation was

0.001705. The 95 percent confidence interval was 0.000431 to 0.007115.

242

Maximum Tensile Strength – HPFRCC Reinforced Torex 2.0%

(a) (b)

(c)


and (c) normal probability plot for HPFRCC reinforced Torex 2.0%



95 percent confidence interval about the mean is 600.95 psi (4.143 MPa) to 1277.15 psi

(8.806 MPa).

The normal probability plot shown in Fig. 7.29 shows that the test data does not



243

Tensile Toughness – HPFRCC Reinforced Torex 2.0%

(a)

(b)

(c)




deviations were 1.24 psi (0.009 MPa). The coefficient of variation was 34.33 percent.

The 95 percent confidence interval was 1.18 psi (0.008 MPa) to 6.04 psi (0.042 MPa).

The normal probability plot shown in Fig. 7.30 shows that the test data does not fit a

straight line, indicating a deviation from normal distribution.

244

Strain at the End of Multiple Cracking Stage – HPFRCC Reinforced Torex 2.0%

(a)

(b)

(c)



Figure 7.31 presents the tensile strain at the end of multiple cracking stage of

HPRCC reinforced Spectra specimen, starting of the localization phase. The means were

0.003993, the corresponding coefficient of variation was 27.98 percent. The standard

deviation was 0.001117. The 95 percent confidence interval was 0.001804 to 0.006182.

245

7.5 Variability Graph

The high variability in strength, strain at maximum stress, and toughness, are

evident in Fig. 7.32 to Fig. 7.34, both of which show the ranges and averages of the

collected data. The upper boundary of each shaded part represents the maximum recorded

strengths, while the lower boundary represents the lowest recorded strengths. The solid

line running through the middle of each shaded region is the average line. The circles at

each Vf represent individual test results at that Vf and the distribution of points again

gives an idea of the large statistical spread in the data.

(a)

(b)

(c)

(d)

Figure 7.32: First cracking stress range and average

246

(a)

(b)

(c)

(d)

Figure 7.33: Maximum stress range and average

247

(a)

(b)

(c)

Figure 7.34: Strain at maximum stress range and average

248

(a)

(b) (c)

Figure 7.35: Energy at maximum stress range and average


The following conclusions are drawn from this study.

1. Generally, a normal distribution curve at 95 percent confidence level is valid for

most tests. However, some tests such as for the maximum stress after cracking

showed varying degrees of departure from normal distribution. This could have

resulted from the relatively large variation in the data.

249

2. The data showed that in general, HPFRCC reinforced Torex has a distribution

nearer to a normal distribution than HPFRCC reinforced with Hooked or Spectra

fibers. HPFRCC reinforced with Spectra fibers showed the largest degrees of

departure from a normal distribution.

3. The increase observed in the tensile strength of different composites is due to the

presence of fibers, and not to the random variation of individual test results. The

difference observed is too great to be attributed to chance.

4. The observed coefficient of variations for the properties of HPFRCC are about 20

percent for maximum tensile strength, 58 percent for strain at maximum stress,

and 48 percent for the corresponding toughness. These variations are larger than

those typically expected for other materials, such as steel, in controlled laboratory

conditions.

5. Variations in the mechanical properties of HPFRCC composites presented in this

study should be considered in deciding the minimum number of tests required in

future tests for measuring material properties, or when selecting the required

material properties for a specified design.

6. There is a direct correlation between the direct tensile test (Dogbone test) results,

and the stress-crack opening displacement test (Notch test) results.

250

CHAPTER 8

RING-TENSILE TEST

8.1 Introduction

A new testing method, the direct ring tensile test, was proposed and discussed in

Chapter 3 as an alternative for the direct tensile test. The advantages of the new test are:

1. More stable compared to the direct tensile test

2. Increased cracking zone. The ring tensile specimen cracking area is significantly

larger than that of the direct tensile test. (Dogbone test) (Fig. 8.1)

3. Ability to scale up specimen size (i.e., easier to increase the dimension in

comparison to Dogbone shaped specimens)

A pilot study was conducted to investigate the appropriate of the method. Four

specimens were employed; ring composites reinforced Hooked 2.0% (Specimen 1 and

Specimen 2), Hooked 1.0%, and specimen reinforced PVA 2.0%.

251

Figure 8.1: Comparison of specimens between ring-tensile and Dogbone shaped

8.2 Measurement Comparisons

(a) (b) (c)

Figure 8.2: Illustrates the displacement obtained (a) from the Optotrak, (b) the LVDT is

on the perimeter of the specimen’s surface, and (c) at the tip of the cone wedge

The test setup is shown in Fig. 8.2 where force is applied to the top of the

specimen’s setup, the cone wedge moves down it pushes out the steel plates. This will

create horizontal force to expand the specimen ring, breaking the specimen. Three types

of sensors were utilized to measure the expansion of the specimen’s perimeter, described

as follows:

252

1. Optotrak system placed on the specimen’s surface (Fig. 8.3)

2. LVDT at the specimen’s perimeter

3. LVDT at the cone wedge tip (bottom part of cone wedge).

A comparison of stress-strain curves obtained from each type of specimen

measurement system is displayed in Fig. 8.4. It is clear from Fig. 8.4 that method 2

measures less displacement at the same stress level than the other 2 systems. Results

showed that the displacement at the cone wedge tip presents the highest error along the

specimen, while displacement measuring by wire at the specimen’s perimeter represents

lower error. Optotrak measurement produces the lowest error along the specimen

measurement.

Figure 8.3: An Optotrak displacement sensor location

(Localization is captured in only half of the specimen)

253

Figure 8.4: Ring-tensile test results of Hooked 2.0% specimen 2 composites with varying

measurements

The localization can be practically captured only from method 3. Optotrak

measurement is pertinent for only half of the specimen. (Fig. 8.3) In the setup, six sensors

were attached, leading to an inaccuracy in measuring the specimen’s perimeter. The other

half is out-of-range and is unable to capture the displacement (Fig. 8.3). If the

localization area is out-of the Optotrak sensors’ range, capturing complete crack-opening

behavior correctly is not possible by this method.

254

Figure 8.5: Ring-tensile test results of Hooked 1.0% composites with various types of

measurements

Figure 8.5 shows a comparison of the three methods for Hooked 1.0% specimens.

From Optotrak sensors, the major localization failure occurs out of Optotrak scopee, but

the internal localization failure occurs in the cracking zone, leads to inability to capture

the displacement after the peak stress. It can be seen clearly that the specimen permanent

deformation is clearly high. Unloading process is clearly visible.

The measurement at perimeter of the specimen is smaller than observed.

However, the measurement at perimeter of the specimen is the smallest. At the cone

wedge displacement, it is too small. The Optotrak sensor is not clearly too small.

Leading to inability of measuring the strain at maximum stress.

When comparing three types of measurements, after calibrating the specimen’s

displacement and calculating the specimen’s strain, method 2 fails to represent the initial

stress-strain data well (i.e., The spring type LVDT is not adequate for the initial stage.)

However, method 1 and 3 provide acceptable data.

255

The localization is very interesting, exhibiting crack formations predominately at

four locations on the specimen’s surface, (e.g., 90 degrees from each other, Fig. 8.13),

and propagated largely when the cone wedge was deeply inserted. After the specimen

reached the rupture point, one of the four of localization zones completely failed (i.e.,

sudden failure; fibers broke).

Figure 8.6: Localization crack of PVA specimen

256

Figure 8.7: Localization crack of Hooked specimen

Figure 8.8: Multiple cracking from inside surface of ring specimen

257

Figure 8.9- One of the four minor localization cracks

Figure 8.10: Multiple cracking for both inside and outside surface of specimens

258

8.3 Testing Results

Figure 8.11: Testing result of all ring specimens tested

Figure 8.11 presents comparison between the stress-strain response of different

specimens: Specimen with Hooked 2.0% specimen-1; Specimen with Hooked 2.0%

specimen-2; Specimen with Hooked 1.0%; specimen with PVA 2.0%. Specimen with

Hooked 2.0% (Specimen 1) gives the highest maximum stress, at 2502 psi (17.26 MPa).

This number is 95.5% higher than found in specimen reinforced Hooked 2.0% specimen-

2 at 1280 psi (8.82 MPa). In specimen reinforced Hooked 1.0%, the maximum stress is

1054 psi (7.26 MPa). Specimen reinforced PVA gives the lowest maximum tensile test,

at 219 psi (1.51 MPa).

259

Figure 8.12: Comparison of maximum stress of all ring-tensile specimens tested

HPFRCC Ring-Tensile testMaximum Stress

2502.6

1279.7

1053.6

218.8

0

500

1000

1500

2000

2500

3000

Hooked 2.0% -1 Hooked 2.0% -2 Hooked 1.0% PVA 2.0%

Max

imum

Str

ess

(ps

260

Specimen’s Cracking Behavior

(a)

Elastic stage - No crack.

(b)

Multiple cracking and minor localization stage. Small cracks were visible. But four minor localization cracks occurred at specimen’s 90 degree corners, satisfying the deflection demand induced from the cone wedge.

(c)

One of the four minor localization cracks became the major localization crack (Largely damaged). The failure was significant and became unstable.

Figure 8.13: Cracking steps in ring-tensile specimen (a) elastic stage, (b) multiple

cracking stage, and (c) localization

261

Comparison Between Ring-Tensile Test and Direct Tensile Test

Test results reveal the difference between the direct tensile test and ring-tensile

test. Overall, it appears that the ring-tensile test gives a higher strength than the direct

tensile test. For example, the maximum resulting from Hooked 2.0% specimen-1, (2502

psi, 17.26 MPa) and Hooked 2.0% specimen-2 (1271 psi, 7.26 MPa) are obtained from

the ring test. However, the result obtained from the direct tensile test of Hooked 2.0% is

only 838 psi (5.78 MPa), and is 1054 psi (7.26 MPa) for the Hooked 1.0% specimen. In

contrast, the average result obtained from the Hooked 1.0% direct tensile is only 535 psi

(3.69 MPa), which is significantly larger.

In addition to strength, the displacement result (such as strain at maximum stress)

differs from the ring-tensile test, generally significantly lower than observations from the

ring-tensile test.

Friction Effect

Almost all specimens, in the study’s pilot simulation process, suffered from

friction. The results from finite element model suggested the value of friction is around

0.3 to 0.5. However, the friction determined from simulation is not effectively applicable

to the testing, due to some inconsistency in the friction coefficient originating from the

manufacturing process of the setup and specimen surface.

262

Figure 8.14: Stress-strain result of PVA 2% specimen with error in strain hardening and

multiple cracking behavior

Testing results reveal the error of stiffness measurement, from Figure 8.14


1. The ring-tensile test system offers for the same volume of material a much larger

area (outer perimeter) than the direct tensile test for observation of cracking,

multiple cracking, and localization. It is also a more stable test than the direct

tensile test.

2. In practice, the Ring tensile test needs further tuning to eliminate or account for

the effect of friction between the cone and steel hedges. Therefore it is not yet

recommended as a replacement for the Direct tensile test.

3. Because of the effect of friction, three to four regions of the tensile ring had

multiple cracking and localization within each region.

263

4. Among the various methods used to measure the circumferential displacement

and thus the strain, the fixture using a thin steel wire attached to an LVDT led to

the most consistent results.

5. The average maximum post-cracking stress observed from the of ring specimens

was higher than that observed from stress-crack opening displacement test.

264

CHAPTER 9

CONCLUSIONS

9.1 Summary

This investigation has provided new insights into the understanding of the

mechanisms for the tensile stress-strain behavior of high-performance discontinuous-fiber

cement-based composites (HPFRCC). Experimental and analytical investigations were

carried out which resulted in quantitative data on the basic properties of fiber reinforced

cementitious composites and models for predicting the post cracking behavior and the

occurrences of multiple cracking in HPFRCC. The experimental data obtained were

maximum stress, strain at maximum stress, energy at maximum stress, energy at the end

of multiple cracking stage, toughness, crack width, crack spacing, strain data, and crack

observations. Predictive models were derived using composite mechanic and energy

principles. The parameters included in the analysis are frictional shear stress, volume

fraction of fiber, matrix cracking stress fiber length, fiber types, as well as fiber diameter.

Analytical models were developed to predict the tensile stress-strain response of fiber

reinforced cement based composites including the elastic stage, multiple cracking stage,

and localization. The study also addressed the variability observed in the tests.

The study consisted of three main phases. The first phase included an extensive

experimental evaluation of the effectiveness of randomly distributed fibers in controlling

tensile behavior of dogbone shaped specimens (direct tensile test). Four types of fibers;

PVA, Spectra, Hooked and Torex, were investigated at volume fractions ranging from

265

0.75% to 2.0%. The second phase studied the influence of randomly distributed fibers on

the responses of notch shaped specimen. The third phase focused on evaluating the

statistical variations in the experimental results. The following conclusions can be drawn

based on the results obtained from the experimental and analytical investigations.

9.2 Conclusions Drawn From Direct Tensile Test (Dogbone) Test of

HPFRCC

9.2.1 PVA Fiber

Testing was carried with two types of otherwise identical fibers: PVA fibers with

oiled surface and PVA fibers with non-oiled surface.

1. Comparing between HPFRCC reinforced with oiled PVA and non-oiled PVA

fibers, the post-cracking strength of specimen with oiled PVA was up to 2 to 3

times that of specimens with non-oiled PVA. Thus the use of oiled PVA fibers

showed significant improvement in tensile behavior, over the short term, as tested

in this study.

2. No clear multiple cracking behavior was observed in specimen reinforced with

PVA fiber regardless of, whether the fibers were oiled or not. By and large, one

crack was observed in all tests. This may seem in conflict with findings from

other investigators, but could be attributed to the fact that they used different

methods of tensile testings and smaller size specimens.

3. The presence of PVA fiber effectively improves the cracking stress, and the

cracking strain. However, comparing improvement with other types of fiber, such

as Spectra, Hooked and Torex, the effectiveness of PVA fiber was the lowest.

4. Increasing the volume fraction of PVA fibers up to 1.5% by volume, leads to a


absorption capacity of the composite.

5. Given the specimen preparation and testing procedure used, the optimum volume

fraction of fiber was close to 1.5%. The highest direct tensile stress observed from

266

specimens reinforced with PVA fibers was 482 psi (3.323 MPa) at 1.5% volume

fraction.

9.2.2 Spectra Fiber

1. The use of Spectra fibers led to a marked improvement in specimen ductility,

energy absorption capacity, strain hardening response and the extent of multiple

cracking. Increasing the volume fraction of fibers up to 1.5% generally led to

improvement in properties. However, at 2% fiber content, properties started

deteriorating due to difficulties in mixing and related air entrapment.

2. Increasing the volume fraction of Spectra fibers up to 1.5% by volume led to an

increase in number of cracks and a decrease in crack width and spacing.

3. A consistent correlation could not be established between the volume fraction of

fiber and the first cracking stress (approximately 300 psi, 2 MPa), the strain at

first cracking, and the corresponding energy absorption capacity.

4. Increasing volume fraction of fiber showed overall improvement in the maximum

post-cracking stress, the strain at maximum stress, and related energy at the end of

multiple cracking or crack saturation.

5. Everything else being equal and given the parameters of this study and the

Spectra fiber used (length and diameter) the optimum volume fraction of Spectra

fiber seems to be around 1.5%.

9.2.3 Hooked Steel Fiber

1. The use of high strength Hooked steel fiber clearly improved the strain hardening

and multiple cracking behavior, toughness, and energy absorption capacity

compared to plain mortar or PVA fiber reinforced specimen. Most specimen

achieved strain hardening response.

2. Increasing the fiber volume fraction clearly led to improvement in all properties

except for the stress and strain at first cracking which did not necessarily follow

the trend.

267

3. Two types of hooked fiber were used, one made with high strength steel wire (of

tensile strength 2100 MPa) and the other with conventional steel wire (of tensile

strength 1050 MPa). Overall the higher strength fiber led to a better performance;

(for instance the maximum tensile stress capacity was 1.5 to 2.5 times at Vf of 1%

to 2%.)

9.2.4 Torex Steel Fiber

1. The use of high strength Torex steel fiber consistently improved the strain

hardening and multiple cracking behavior, the maximum post-cracking stress, the

strain and energy at maximum stress, and related energy absorption capacity. The

higher the volume content of fibers, the higher the improvements observed in the

tensile behavior.

2. The number of cracks observed generally increased with the volume fraction of

fiber, leading to a decrease in crack spacing and width. Specimens with high

volume fraction of fiber also showed large ductility and energy absorption

capacity.

3. Two types of Torex fiber were used in this study, one made with high strength

steel wire (of tensile strength 2760 MPa), and one with normal strength steel wire

(of tensile strength 1380 MPa). The higher strength fiber led to a 30% to 60 %

better performance, in terms of maximum stress.

9.2.5 Direct Tensile Tests: General Conclusions

1. Generally, the first cracking stress of the composite is significantly improved due

to the presence of fiber in the matrix. However, no consistent correlation could be

established between an increase in volume fraction of fiber and an increase in first

cracking stress. Given the parameters of this study, for Hooked and Torex fiber,

the stress at first cracking increased with the volume fraction, while it remained

almost same for PVA and Spectra fibers

268

2. Comparing between types of fiber, for the same volume fraction of fibers,

specimens with Torex, high strength Hooked, and Spectra fibers showed better

overall behavior than specimens with PVA fibers. Moreover, their post-cracking

strength reached 1.5 to 3 times their strength at first cracking, while for specimens

with PVA fibers, the post-cracking strength was only 10% to 50% higher than the

cracking strength at up to 2% fiber content.

3. The tensile response of specimens without fibers is very brittle and shows large

variability in tensile strength. Fibers, whether leading to strain softening or strain

hardening response, reduce the variability while improving ductility and

toughness.

4. Immediately following localization, specimens reinforced with PVA fiber fail

suddenly due to tensile failure of the fibers. For specimens reinforced with

Spectra fiber, the failure is gradual and controlled by the pulling out of the fibers

accompanied by matrix spalling around the critical crack. Also, for specimens

reinforced with steel fiber, whether Hooked or Twisted, the fibers gradually pull

out up to complete separation; during pull out, twisted fibers untwist leading to

additional matrix cracking. No steel fiber failure was observed for the variables

of this study.

5. For the same volume fraction, and type of steel fiber, the strength of fiber is

important for determining the composite tensile behavior. Specimen reinforced

with high strength steel fiber usually outperform specimen reinforced with regular

strength steel fiber.

6. Changing surface properties of the fiber has important implications. For instance,

specimens reinforced with oiled PVA fibers performed better than specimens

reinforced with non-oiled ones.

7. Variability of properties obtained from direct tensile testing is large and a fact that

cannot be ignored. Both fiber distribution within the specimen and fiber

orientation at any section play a significant role in influencing observed

properties. The more uniform is the distribution of fiber, the less variable the

tensile behavior is.

269

9.3 Conclusions Drawn from the Stress-Crack Opening Displacement

(COD) Tests on Notched Tensile Prisms

9.3.1 PVA Fiber

1 No multiple cracking was observed in these specimens at all volume fractions

tested. However, strain hardening behavior was observed in specimens with high

volume fractions of PVA fiber. Such behavior could be attributed to the slow

propagation of a single crack from one end of the section to its other end.

2. Only single cracks were observed in these tests. Each crack was fine and well

defined with no large damage zone around it .

3. For the given parameters of this study, the optimum volume fraction of fiber is

1.5%, the same as observed from the direct tensile tests. However, the average

maximum stress of the notched prisms (about 3.9 MPa) was higher than that

obtained from the dogbone tensile prisms (about 3.7 MPa).

4. Comparing the displacement at maximum stress between specimens with high

strength Hooked, Torex, Spectra, and PVA fiber, the smallest displacement was

by far with the PVA fiber. Typically such displacement was 10 times smaller than

observed from specimens with the other fibers.

9.3.2 Spectra Fiber

1. Multiple cracking behavior around the notched section (equivalent to smeared

cracking) was clearly observed. Such behavior encourages strain hardening (or

high performance) response in direct tension as indeed observed in the direct

tensile tests.

2. The cracking zone on either side of the notch area (area between two notches) is

significant, containing several cracks. Its extent increased with an increase in

volume fraction of fibers. Specimens containing high volume fraction of fibers

usually outperformed those containing low volume fraction of fibers.

270

3. No consistent correlation could be established between the fiber content and the

first cracking stress, the strain at first cracking stress, and the energy at first

cracking stress. These variables seemed independent of the volume fraction of

fibers for the range of fiber content tested. However, the maximum stress, strain

at maximum stress, and energy at the end of multiple cracking stage all increased

with an increase in volume fraction of fiber.

4. Keeping in mind mixing difficulty and air entrapment, the best performance was

obtained at a volume fraction of 1.5% (not 2%) and is consistent with the results

from the direct tensile (dogbone) tests.

9.3.3 Hooked Steel Fiber

1. Numerous multiple cracks clearly occurred around the notched section.

Displacement (strain) hardening behavior was observed. The higher the volume

fraction of fibers, the more extended was the cracked region. Specimens with

2.0% and 1.5% volume fractions were observed as having a larger number of

crack (about 5 equivalent cracks).

2. Generally, the higher the volume fraction of fibers the better the post-cracking

strength and ductility. However, the displacement at maximum stress was almost

independent of the fiber volume fraction.

3. The stress at first cracking and related energy increased slightly with the volume

fraction of fibers. However, the strain at first cracking stress remained almost

constant.

4. A high variability in test results was clearly observed.

9.3.4 Torex Steel Fiber

1. The use of Torex fibers led to significant multiple cracking around the notched

section, and displacement (or strain) hardening behavior was observed.

Specimens reinforced with Torex fiber achieve high performance behavior. The

271

strain hardening and multiple cracking behavior, with significant ductility and

energy absorption, are clearly observed.

2. Multiple cracking was observed even at 0.75% fiber content. The higher the

volume fraction of fibers, the higher the maximum post-cracking stress and

related energy absorption capacity, and the larger the number of equivalent

cracks. Everything else being equal, maximum stresses here exceeded those

obtained from direct tensile tests.

3. The stress at first cracking and related strain did not seem to depend on the fiber

volume fraction.

4. The damaged area around the notched section was usually smaller than that using

Spectra fibers. This could be attributed to the fact that the Spectra fibers used

were 38 mm in length compared to the 30 mm Torex fibers.

5. The variability in test results obtained with Torex fibers was smaller than that

with Spectra and Hooked fibers.

9.3.5 Stress Crack Opening Displacement (COD) Tests: General Conclusions

In this study only one mortar matrix composition and strength was used. It is likely

that results would be different with different strength matrices. However, it is also likely

that typical observation, for the shape of the stress versus COD curves, will be similar to

what has been observed in this study.

The following conclusions can be drawn from this study:


classified into four types : clearly strain hardening with multiple cracks ; clearly

strain softening with a single localized crack ; and two cases in between, a strain

hardening material with a single major crack ; a strain softening material with

post cracking strength, able to pick up almost up to the cracking stress.

2. For the same volume fraction of fibers, specimens with Torex and High strength

Hooked fibers showed better overall behavior than specimens with Spectra and

PVA fibers. Moreover, their post-cracking strength was 1.5 to 3 times the strength

272

at first cracking, while for specimens with PVA fibers, the post-cracking strength

was only 10% to 50% higher than the cracking strength.

3. For the same type of fiber, increasing the volume fraction led to a marked

improvement in post-cracking strength, ductility, and energy absorption capacity

of the composites.

4. The crack opening in specimens with Torex, high strength Hooked and Spectra

are the result of fiber-pulling out. However, the damage in specimens with PVA

fibers was due to fiber breaking.

9.4 Conclusions from the Study on Modeling Tensile Stress-Strain

Response

A model based on composite mechanics and experimental observations, to explain the

occurrence of multiple cracking in discontinuous fiber composites, (HPFRCC) was

developed. The influence of various parameters was investigated. Good agreement was

found between the model predictions for stress-strain relation and experimental results.

The following specific conclusions are drawn.

1. Predictive equations derived from mechanics of composites were used for the

stress at first cracking and the maximum post-cracking stress. From the

experimental results obtained, prediction equations were suggested for the

coefficients in these equations,

2. Predictions equations for average crack spacing and average crack width for

HPFRCC tensile specimens reinforced with Spectra, Hooked, and Torex fibers

were derived based on experimental observations.

3. A method to predict the strain at maximum stress, from crack width and crack

spacing is suggested. Good agreement between predicted and experimental results

was observed.

4. A analytical equation was developed to predict the response of the composite in

tension after localization. The equation depends on two parameters only, and

allows to simulate damage as well as variable curvature in the shape of the curve.

273

The model produced good approximation of the localization behavior obtained

experimentally.

5. A correlation between the direct tensile test (Dogbone test) and stress-crack

opening displacement test (Notch test) was established. The proposed method was

found to have a good agreement between prediction and experiment.

9.5 Conclusions from the Statistical Analysis and Variability of Results

1. Generally, a normal distribution curve at 95 percent confidence level is valid for

most test results. However, some tests such as for the maximum stress after

cracking showed varying degrees of departure from normal distribution. This

could have resulted from the relatively large variation in the data.

2. The data showed, generally, HPFRCC reinforced Torex has a distribution nearer

to the theoretical normal distribution, better when compared to HPFRCC

reinforced with Hooked and Spectra fibers. HPFRCC reinforced with Spectra

fibers showed the largest degrees of departure from normal distribution.

3. The increase observed in the tensile strength results of different composites is due

to the presence of fibers, and not to the random variation of the individual test

results. The difference observed is too great to be attributed to chance.

4. The observed coefficient of variation for the properties of all HPFRCC testes,

were about 20 percent for maximum tensile strength, 58 percent for strain at

maximum stress, and 48 percent for the corresponding toughness. These

variations are larger than those typically expected for other materials, such as

steel, in controlled laboratory conditions.

5. Variations in the mechanical properties of HPFRCC composites presented in this

study should be considered in deciding the minimum number of tests required in

future tests for measuring material properties, or when selecting the required

material properties for a specified design.

6. There is a direct correlation between the direct tensile test (Dogbone test) results,

and the stress-crack opening displacement test (Notch test) results.

274

9.6 Conclusions for the Ring Tensile Test Study

The following conclusions are drawn from this part of the study.

1. The principle, the ring-tensile test system offers for the same volume of material a

much larger area (outer perimeter) than the direct tensile test for observation of

cracking, multiple cracking, and localization. It is also a more stable test than the

direct tensile test

2. In practice, the Ring tensile test needs further tuning to eliminate the effect of

friction between the cone and steel ring during testing. Therefore it is not yet

recommended as a replacement for the Direct tensile test.

3. Because of friction, three to four regions of the tensile ring had multiple cracking

and localization within each region.

4. Among the various methods used to measure the circumferential displacement

and thus the strain, the fixture using a thin steel wire attached to an LVDT led to

the most consistent results.

5. The average maximum post-cracking stress observed from the of ring specimens

was higher than that observed from stress-crack opening displacement test.

9.7 Main Conclusions

1. The first cracking stress of the composite is significantly improved due to the

presence of fibers in the matrix. However, the correlation between volume

fraction and first cracking stress, in some cases, depends on volume fraction of

fibers (such as for Hooked and Torex fibers) and in other cases seems

independent of the volume fraction of fibers (such as for PVA and Spectra fibers.)

2. Immediately following localization, specimens with PVA fiber failed by failure of

the fiber. For the other fibers used (Spectra, Hooked and Torex) failure occurred

by fibers primarily pulling out of the matrix. With Spectra fibers, significant

spalling of the matrix occurs during the softening response.

275

3. Comparing tensile performance of the composites at the same fiber volume

fraction, Torex fibers perform the best, Hooked and Spectra fibers are next, and

PVA fibers give the lowest tensile performance of all fibers tested.

4. For the same type of fiber, increasing the volume fraction generally leads to a


absorption capacity of the composites.


classified into four types : clearly strain (or displacement) hardening with multiple

cracks ; clearly strain (or displacement) softening with a single localized crack ;

and two cases in between, a strain hardening material with a single major crack ;

and a strain softening material with post cracking strength, able to pick up almost

up to the cracking stress.

6. Unlike what could be anticipated, stress versus crack opening displacement tests,

using notched tensile prims, generate multiple cracks or a cluster of cracks in the

zone of influence surrounding the notched section. Such behavior was observed

in all specimens reinforced with Hooked, Spectra and Torex fibers. Therefore the

notion of observing only one crack such as in the case of PVA fiber should be

carefully revised and accommodated in any modeling.

7. The strain at maximum post-cracking stress seems to be independent of fiber

volume fraction for hooked fibers, but increases slightly with the fiber volume

fraction for Torex fibers.

8. The experimental results combined with a simple composite mechanics approach

allowed for a rational prediction of the key properties of HPFRCC in tension

leading to simple prediction equations of these properties. This allowed for

predicting schematic simplified stress-strain and stress-displacement response

curves for use in structural modeling.

9. A large variability should be expected from the tensile properties of fiber

reinforced cement composites, and this should be accounted for in design. Some

effort should be expanded in the future to study the causes of such variability and

attempt to reduce it.

276

9.8 Recommendations for Future Work

An extensive experimental program has been carried out in this research to

understand the response of HPFRCC post cracking characteristic. However, the following

research works are recommended.

1. Measurement of elastic bond strength, frictional bond strength, and debonding

energy will provide exact information on interface properties which will be

needed for the application of the model. Different bonding condition can be

simulated by using fiber with different coating materials.

2. Significant additional research is needed to evaluate size effect on tensile

properties of HPFRCCs, particularly the strain at maximum post-cracking

stress and the multiple cracking process.

3. Additional work is needed to fine tune the ring test and eliminate the effect of

friction between the moving cone and the ring.

4. Additional experimental tests are needed on the the notched tensile prisms

used for the stress versus crack opening displacement, particularly to make

sure that cracking remains within the notched section (such as by using larger

notches).

5. Only the tensile response of composites was investigated in this research.

More experimental work is needed to evaluate the composite response in other

types of loading such as biaxial loading (tension-tension or tension

compression), shear, cyclic loading, etc.

277

APPENDICES

278

APPENDIX A

DIRECT TENSILE TESTS USING SIFCON

Five SIFCON (Slurry Infiltrated Fiber Concrete) Dogbone specimens reinforced

with high strength Torex steel fiber and high strength Hooked steel fiber were tested to

study the behavior of composites at very high volume fraction, (5% for the series with

Hooked fiber and 4% for the series with Torex fiber). The results are illustrated in Fig. A-

1 and Fig. A-2. Note that the volume fraction of fiber is obtained from the difference in

the weight of the molds before and after placing the fibers.

Figure A-1: Stress-Strain curves of

specimens reinforced with Hooked fiber Figure A-2: Stress-Strain curves of

specimens reinforced with Torex fiber

279

Figure A-3: Comparison of Average stress-strain curves

From the results it can be observed that SIFCON specimens, not only achieve

much higher strength, but also larger ductility and energy absorption (area under the

curves at the end of multiple crack stage (15.0 psi for series with Torex and 7.4 psi for

series with Hooked) than observed from the series with 2% volume fraction of fiber (3.6

psi for series with Torex and 2.92 psi for series with Hooked). Further, their strain at

maximum stress is much higher They also developed a extensive multiple cracking with

very fine crack width and small crack spacing.

280

Figure A-4: Preparation of SIFCON specimens (mold and

fibers)

Figure A-5: SIFCON surface with large number of fiber in the

specimens. (Torex)

Figure A-6: SIFCON

specimen after testing.

281

Table A-1: Summary of test results of SIFCON (US-units)

Table A-2: Summary of test results of SIFCON (US-units)

282

APPENDIX B

DEFINITION OF FIRST CRACKING, MAXIMUM STRESS POINT, AND LOCALIZATION STARTING POINT

(a)

(b)

Figure B-1: Definition of first cracking point, maximum stress point, and localization

starting point at (a) strain (displacement) up to 0.02 and (b) strain (displacement) up to

0.006

First cracking point: The first cracking point refer to the point where the elastic

stiffness is significantly changed. It is also the point where multiple cracking starts.

Maximum stress point: The maximum stress point refer to the point at which the

specimen reached maximum stress

Localization starting point: The localization starting point is the end of multiple

cracking point. It can be defined as the point at which the descending stiffness

undergoes a significant drop. Usually at this point, the stress level is about between the

maximum stress and 75% of the maximum stress. 75% to 100% of maximum stress.

283

Figure B-2: Definition of maximum stress point, and localization starting point for a

specimen with Spectra fiber

Figure B-2 shows the stress-strain

response of HPFRCC reinforced with 2%

Spectra fiber. Here, clearly the maximum

stress point and localization starting point

are not the same. The maximum stress

point occurs at around 0.005 strain, but the

localization starting point occurs at around

0.02 strain. Therefore, the localization

starting point is not the maximum stress

point.

Figure B1-a and B1-b and Fig. B2 illustrates various examples of these points.

284

APPENDIX C

VARIBILITY OBSERVED IN SAME TEST SERIES USING 12 SPECIMENS PREPARED ON THE SAME DAY AND TESTED ON

THE SAME DAY

Figure C-1: Stress-strain curves of Series

with Torex 1.5%

Figure C-2: Stress-strain curves of Series

with Torex 2.0%

Twenty four tensile specimens (Dogbone) were tested in direct tension similarly

as described in Chapter 4. Twelve specimens contained 1.5% Torex fiber by volume and

the other twelve contained 2% Torex fibers by volume. These two series were cast on the

same day and tested on the same day. Thus their variability should be representative for

such conditions. The stress-strain curves are plotted in Figs. C1 and C2 and presents

significantly low variability than observed from the series presented in Chapter 4 where

up to 29 specimens prepared and tested at different time were analysed. Table C-1

summarizes the main results. . It can be observed that the coefficients of variation of the

the maximum stress, maximum strain, and corresponding toughness are significantly

smaller than reported in Chapter 4.

285

Table C-1: Summary of comparison of coefficient of variation ,COV (%)

COV (%) Torex

Volume fraction

Series comparison Maximum Stress

Strain at Maximum

Stress

Toughness (Energy)

Mixed and tested at different time. 23.59 51.36 51.38 1.50% Mixed the same time, Tested the same date 11.96 24.89 35.02

Mixed and tested at different time. 18.39 27.97 34.33 2.00%

Mixed the same time, Tested the same date 17.48 15.93 31.3

Table C-2: Summary of test results of series reinforced Torex 1.5%(US-units)

Table C-3: Summary of test results of series reinforced Torex 1.5%(SI-units)

286

Table C-4: Summary of test results of series reinforced Torex 2.0%(US-units)

Table C-5: Summary of test results of series reinforced Torex 2.0%(SI-units)

287

APPENDIX D

COMPARISON OF α1α2, α1α2τ, λpc AND λpcτ

In order to estimate the coefficients α1α2 and λpc

(Chapter 6) the bond strength τ

was assumed for each type of fiber. Since the value of bond strength can be subjective,

another way can be to consider τ unknown, and estimate from the data the values of α1α2τ

and λpcτ. The results are summarized in Table D1. It can be observed that the fiber

contribution to the postcracking strength, λpcτ, generally decreases with an increase in the

volume fraction of fibers. For the contribution at onset of cracking, the product α1α2τ

does not show any clear trend; however, given the variability observed in the results, it

can ba assumed constant for a fiber type.

288

Figure D-1: Comparison of α1α2 and α1α2τ

Figure D-2: Comparison of λpc and λpcτ

289

BIBLIOGRAPHY

290

BIBLIOGRAPHY

1. Abrishami, H.H., and Mitchell, D., 1996, “Influence of Splitting Cracks on

Tension Stiffening,” ACI Structural Journal, V. 93, No. 6, Nov-Dec,. pp.703-710.

2. Abrishami, H. H., and Mitchell, D., 1997, “Influence of steel fibers on Tension

Stiffening,” ACI Structural Journal, V. 94 No. 6, Nov-Dec., pp. 769-776

3. ACI Committee 224, 1992, “Cracking of Concrete Members in Direct Tension

(ACI 224.2R-92),” American Concrete Institute, Famington Hills, Mich., 12pp,

4. Akkaya, Shah and Ankenman., 2001, “Effect of Fiber Dispersion on Multiple

cracking of Cement Composites,” Journal of Engineering Mechanics, April

2001,”

5. Alwan, Jamil M. (1994) “Modeling of the mechanical behavior of fiber reinforced

cement based composites under tensile loads” Ph.D. Dissertation, University of

Michigan, Ann Arbor

6. A.N. Lambrechts, “The variation of steel fiber concrete characteristic. Study on

toughness results 2002-2003”, N.V. Bekart S.A., Belgium, Fiber Reinforced

Concrete From Theory to Practice, International Workshop on Advances in Fiber

Reinforced Concrete, Bergamo, Italy, September 24-25, 2004

7. Aveston et al. (1971) “ Single and multiple fracture In: The properties of fiber

composites” Conference Proc. National Physical Laboratory, IPC Science and

Technology Press Ltd., 15-24

8. ASTM C1018-94b, “Standard Test Method for Flexural Toughness and First-

Crack Strength of FRC, ASTM Committee on Standards, Philadelphia, 1994.

9. Balaguru, P.N, Shah, S.P, “Fiber-Reinforced Cement Composites”, McGraw-Hill

International Editions, 1992

10. Chandrangsu, K and A.E. Naaman, Comparison of Tensile and Bending response

291

of three high performance fiber reinforced cement composites, HPFRCC4

Workshop, Ann Arbor, USA, 2003

11. Chandrangsu, K, “Innovative Bridge Deck with Reduced Reinforcedment and

Strain-Hardening Fiber Reinforced Cementitious Composites,” Ph.D. thesis,

University of Michigan, Ann Arbor, 2003

12. Cook, Robert, D; David S. Malkus; Michael E. Plesha and Robert J. Witt,

“Concepts and Applications of Finite Element analysis”, Fourth Edition; Wiley

13. Cooper G.A. and J.M. Sillworrd. 1972, “Multiple fracture in a Steel Reinforced

Epoxy Resin Composite.” J of American Ceramic Society 75(2) pp 316:324

14. Dare, P.M., Hanley, H.B., Fraser, C.S., Riedel, B. & Niemeier, W. (2002), “An

Operational Application of Automatic Feature Extraction: The Measurement of

Cracks in Concrete Structures”, Photogrammetric Record, 17, 1999, pp 453-464

15. Dhanada K. Mishra (1995), “Design of Pseudo Strain-Hardening Cementitious

Composites for a Ductile Plastic Hinge”, Ph.D. Dissertation, University of

Michigan, Ann Arbor.

16. Fischer, Gregor and Li, Victor C., 2002, “Influence of Matrix Ductility on

Tension-Stiffening Behavior of Steel Reinforced Engineered Cementitous

Composites (ECC),” ACI Structural Journal, Jan-Feb , pp104-111

17. Guedes, J.M., “Nonlinear Computational Models For Composite Materials Using

Homogenization”, Ph.D. Thesis, Department of Applied Mechanics, University of

Michigan, 1990

18. Guerrero Z. (1999) ‘Bond stress-slip mechanisms in high performance fiber

reinforced cement composites” Ph.D. Dissertation, University of Michigan, Ann

Arbor.

19. Hansen, W., “Tensile Strain Hardening in Fiber Reinforced Cement Composites”,

Proc. Of the International RILEM/ACI Workshop, High Performance Fiber

Reinforced Cement Composites, June 1991 pp 419-428

20. Hauwaert, A.V. ; Delannay, F and Thimus, J-F, “Cracking behavior of Steel Fiber

Reinforced Concrete Revealed by Mans of Acoustic Emission and Ultrasonic

Wave Propagation”, ACI Materials Journal, May-June 1999, pp291-296

292

21. Hong-Gyoo Sohn, Yun Mook Lim, Kong-Hyun Yun & Gi-Hong Kim,

Monitoring Crack Changes in Concrete Structures, Computer-Aided Civil and

Infrastructure Engineering 20 (2005) pp52-61

22. Homrich and Naaman, (1987) “Stress-Strain Properties of SIFCON in Uniaxial

Compression and Tension”, Department of Civil Engineering, Unpublished.

23. Kabele (2004) “Linking scales in modeling of fracture in high performance fiber

reinforced cementitious composites” Fracture Mechanics of Concrete Structures,

Ia-FraMCos, 71-80

24. Kanda, T and Li, V.C., “New Micromechanics Design Theory for Pseudostrain

Hardening cementitious Composite”, Journal of Engineering Mechanics, April

1999, pp 373-381

25. Kanda, T.; Lin, Z.; Li, V.C., “Tensile Stress-Strain Modeling of Pseudostrain

Hardening Cementitious Composites”, Journal of Materials in Civil Engineering,

May 2000, pp 147-156

26. Karihaloo B.L. and Wang, J, “Mechanics of Fiber-reinforced Cementitious

Composites”, Computer & Structures, 76, 2000 (19-34)

27. Kimber, A.C. and J.G. Keer. 1982. “On the Theoretic Average Crack Spacing in

Brittle Matrix Composites Containing Continuous Aligned Fibers,” J of Material

Science letter., 1:353:354

28. Kosa, Kenji and A.E. Naaman, “Corrosion of Steel Fiber Reinforced Concrete”,

ACI materials Journal, V.2, No.1, January – February 1990, pp.27-37

29. Kullaa, J ,”Micromechanics of multiple cracking Part II Statistical tensile

behavior”, Journal of Materials Science 33 (1998)

30. Leonhardt, F., “Crack Control in Concrete Structures,” IABSE Surveys No. S-

4/77, International Association for Bridge and Structural Engineering, Zurich,

1977, 26pp

31. Leung and Li (1991) “New strength based model for the debonding of

discontinuous fibers in an elastic matrix” Journal of Material Science, 26, 5996-

6010

293

32. Li, V.C., Wu, H. C., and Chan, Y.W. (1996), “Effect of plasma treatment of

polyethylene fibers on interface and cementitious composite properties”, J. Am.

Ceramics Soc., 79(3), 700-704

33. Li, V.C., Lin, Z. and Matsumoto, T., “Influence of Fiber Bridging on Structural

Size-Effect”, Int Solids Structures, Vol 35 No 31 1998, pp 4223-4238

34. Li, V.C.; Wang, S and Wu, C, “Tensile Strain-Hardening Behavior of Polyvinyl

Alcohol Engineered Cementitious Composite (PVA-ECC), ACI Material Journal,

November-December 2001, pp 483-492

35. Li, V.C.; Wu, Hwai-Chung.; Maalej, M and Mishra, D. K., “Tensile Behavior of

Cement-Based Composites with Random Discontinuous Steel Fibers”, Journal of

the American Ceramic Society, January 1996, pp 74-78

36. Li, V.C.; “From Micromechanics to Structural Engineering, The Design of

Cementitious Composites for Civil Engineering Applications”, Structural

Engineering/ Earthquake Engineering Vol 10, no 2 July 1993, pp 37-48

37. Lin, Zhong and Li, V.C., “Crack Bridging in Fiber Reinforced Cementitious

Composites with Slip-Hardening Interfaces”, Journal of Mec and Phys Solids,

1997, pp763-787

38. Matthews, F.L. and Rawlings, R.D., “Composite Materials: Engineering and

Science”, Woodhead Publishing, CRC Press, 470 pages.

39. Mitchell, Denis and Collins, Michael, P, “Prestressed Concrete Structures”,

Prentice Hall,1990

40. Mitchell, Denis: Hosseini, Homayoun and Mindess, Sidney, “The Effect of Steel

Fibers and Epoxy-Coated Reinforcement on Tension Stiffening and Cracking of

Reinforced Concrete”, ACI Materials Journal, January-February 1996, pp.61-67

41. Mobasher, B, “Recent Advances in Modeling fiber Reinforced Concrete”, Fiber

Reinforced Concrete From Theory to Practice, International Workshop on

Advances in Fiber Reinforced Concrete, Bergamo, Italy, September 24-25, 2004.

42. Murugappan, K., Paramasivan, P., and Tan, K. H., “Shear Response of

Reinforced Fibrous Concrete Beams Using Finite-Element Method”, Fiber

Reinforced Cement and Concrete, RILEM, 1992, pp 447-463

43. Naaman, A.E, “Ferrocement & Laminated Cementitious Composites,” Techno

294

Press 3000, Ann Arbor, 2000, 372 Pages.

44. Naaman, A.E.; Moavenzadeh, F and McGarry, F., “Probabilistic Analysis of Fiber

Reinforced Concerete, “Journal of Engineering Mechanics Division, ASCE, V.

100, No. EM2, Apr. 1974, pp.397-413

45. Naaman, A.E. and Reinhardt, H.W., “Characterization of High Performance Fiber

Reinforced Cement Composites,” Proceedings of 2nd International Workshop on

HPFRCC, Chapter 41, in High Performance Fiber Reinforced Cement

Composites: HPFRCC 2, RILEM, No 31, 1996, pp. 1-24.

46. Naaman, A.E. and H.W. Reinhardt, ,”Setting the Stage: Toward performance

based classification of FRC Composites,” High Performance Fiber Reinforced

Cement Composites (HPFRCC4), Workshop, Ann Arbor, USA 2003

47. Naaman, A.E., Moavenzadeh, F and McGarry, F.J., “Probabilistic Analysis of

Fiber-Reinforced Concrete”, ASCE Engineering Mechanics Division, April 1974

48. Naaman, A.E., and Homrich, J.R., “Tensile Stress-Strain Properties of SIFCON”,

ACI Materials Journal, May-June 1989, pp 244-251

49. Naaman, A.E., Otter D., and Najm H. (1991), “Elastic Modulus of SIFCON in

Tension and Compression., “ACI Materials Journal, Vol. 88, No.6, pp.603-612

50. Naaman, A.E. and Shah, S.P., “Fracture and Multiple Cracking of Cementitious

Composites”, Fracture Mechanics Applied To Brittle Materials, J.W. Freiman,

ed., ASTM, 1979, pp 183-201

51. Najm, H., and Naaman, A.E., “Prediction Model for the Elastic Modulus of High

Performance Fiber Reinforced Cement Based Composites, “ACI Material Journal,

May-June 1995, pp 304-314

52. Nammur, G.G., “Characterization of Bond in Steel Fiber Reinforced Cementitious

Composites under Tensile Loads”, The University of Michigan, 1989

53. Nelson, P, K; Li, V.C. and Kamada, T, “Fracture Toughness of Microfiber

Reinforced Cement Composites”, Journal of Materials in Civil Engineering,

September-October 2002, pp 384-391

54. Ortlepp, Regine; Hampel, Uwe and Curbach, Manfred;, “A New Approach for

Evaluating Bond Capacity of TRC Strengthening, Dresden University of

Technology, Department of Civil Engineering, 2005, Unpublished.

295

55. Peled, Alva and Mobasher, Barzin, “Pultruded Fabric-Cement Composites”, ACI

Materials Journal, January-February 2005, pp15-23

56. Romualdi, J.P., and Mandel, J.A., “Tensile Strength of Concrete Affected by

Uniformly Distributed and Closely Spaced Short Lengths of Wire

Reinforcement”, ACI Journal, Proceedings, vol 61, no.6. June 1964, pp 657-670

57. Rossi, Pierre, “Steel Fiber Reinforced Concretes (SFRC): An Example of French

Research”, ACI Material Journal, May-June 1994, pp273-279

58. Rossi, P. and Richer, S., “Numerical Modelling of Concrete Cracking Based on a

Stochastic Approach”, Materials and Structures”, 1987, pp 334-337

59. Rossi, P., and Wu, X., “Dimensioning and Numerical modeling of Metal Fiber

Reinfroced Concrete (MFRC) Structures”, Cement and Concrete Composites,

1992, pp. 195-198

60. Schutter, G., “Advanced Monitoring of Cracked Structures Using Video

Microscope and Automated Image Analysis”, NDT&E International, 35(4), 2002,

pp 209-212

61. Sirijaroonchai, Kittinun, “Modeling of High Performance Fiber-Reinforced

Cement Composite Structures, Ph-D Proposal, University of Michgian, 2005,

Unpublished.

62. Sujiravorakul, C., “Development of High Performance Fiber Reinforced Cement

Composites Using Twisted Polygonal Steel Fibers,” Ph.D. thesis, University of

Michigan, Ann Arbor, February 2001

63. Suwannakarn, S., Zhu, Z. and Brilakis, I. (2007) "Automated Air Pockets

Detection for Architectural Concrete Inspection", ASCE Construction Research

Congress, 6-8 May 2007, Freeport, Bahamas

64. Suwannakarn, S., El-Tawil, S., and Naaman, AE., “Experimental Observations on

the Tensile Response of Fiber Reinforced Cement Composites with Different

Fibers”, High Performance Fiber Reinforced Cement Composites (HPFRCC5)

Mainz, Germany – July 10-13, 2007 pp39-47

65. Suwannakarn, Naaman, AE., and El-Tawil, S., “ Stress Versus Crack Opening

Displacement Response of FRC Composites with Different Fibers”, Seventh Intnl.

In RILEM Proceedings PRO 60, "Fiber Reinforced Concrete: Design and

296

Applications,", Edited by Ravindra Gettu, Rilem Publications, France, 2008, pp.

1039-1054.

66. Swamy, R.N., Mangat, P.S., and Rao, C.V.S.K., “The Mechanics of Fiber

Reinforcement of Cement Matrices”, An International Symposium: Fiber

Reinforced Concrete, Publication SP-44, ACI, 1974, pp 1-28

67. Vecchio, F.J., and Collins, M.P., “Modified Compression Field Theory for

Reinforced Concrete Elements Subjected to Shear,” ACI Structural Journal, V.83,

No.2 Mar-Apr 1986, pp 219-231

68. Vellore S. Goapalaratnam and Surendra P. Shah, “Tensile Failure of Steel Fiber-

Reinforced Mortar”, Journal of Engineering Mechanics, No 5, May 1987

69. Visalvanich, K and A.E. Naaman, “Fracture Model for Fiber Reinforced

Concrete”, ACI Journal, March-April 1983

70. Vellore S. Goapalaratnam and Surendra P. Shah, “Tensile Failure of Steel Fiber-

Reinforced Mortar”, Journal of Engineering Mechanics, No 5, May 1987

71. Tjiptobroto, Prijatmadi, “Tensile Strain-hardening of High Performance Fiber

Reinforced Cement-Based Composites”, Ph-D thesis, University of Michigan,

Ann Arbor, 1991

72. Tjiptobroto, Prijatmadi and Hansen, Will, “Tensile Strain-hardening and Multiple

Cracking in High-Performance Cement-Based Composites Containing

Discontinuous Fibers”, ACI Materials Journal, January-February 1993.

73. Tung, P. C., Hwang, Y.R. & Wu, M.C., “The Development of a Mobile

Manipulator Imaging System for Bridge Crack Inspection”, Automation in

Construction, 11(6), 2002, pp 717-729

74. Wongtanasitcharoen, Thanasak, “A Study of The use of Fibers to Control Plastic

Shrinkage Cracking of Concrete”, Ph-D Proposal, University of Michgian, 2002,

Unpublished.

75. Wongtanasitcharoen, Thanasak, “Effect of Randomly Distributed Fibers on

Plastic Shrinkage Cracking of Cement Composites”, Ph-D thesis, University of

Michigan, Ann Arbor, 2005

76. Wu, H..,C, T. Matsumoto and V.C.Li. 1994, “Buckling of Bridging Fibers in

Compsotes.” , J. Materials Science Letters, Vol. 13, pp. 1800-1803, 1994.

297

77. Yan, Wu and Zhang, “A quantitative study on the surface crack pattern of

concrete with high content of steel fiber”, Cement and concrete research, 21

(2002).

78. Yang and Fisher, “Investigation of the Fiber Bridging Stress-Crack Opening

Relationship of Fiber Reinforced Cementitious Composites”, HPFRCC

Conference 5, 2006

79. Yang X.F. and K.M. Knowles. 1992. “The One Dimensional Car Parking

Problem and its Application to the Distribution of Spacing between Matrix Cracks

in Unidirectional Fiber-Reinforced Brittle Materials.” Journal of American

Ceramic 75 pp 141-147

80. Zeng W., Kwan, A.K.H. and Lee, P.K.K., “Direct Tension Test of Concrete”, ACI

Materials Journal, January-February 2001 (63-71)

post-cracking characteristics of high performance fiber reinforced

Documents

Transcript of post-cracking characteristics of high performance fiber reinforced